Chuyên đề Hình học Không gian lớp 12 - Lê Trường Sơn

 12 CI
 Trang
1

: THỂ TÍCH LĂNG TRỤ


1) Dạng 1:     có  cao hay  

 

2

 

 
 
 

này.
  60
0
. 


* Bài tập tương tự:
Bài 1:  a.
 
3
a3
V

4

; S = 3a
2

Bài 2: 
BD' a 6
.
 
3

Bài 3: 
  
 
3
và S = 248cm
2

Bài 4:  

2
 
3

Bài 5: 


3

Bài 6: 


2
 
3

Bài 7: 
 
Bài 8: 
2




Bài 9: 
 
3

Bài 10: 
5; 10; 13
.
 
Bài 11: 
0
60C 
. 
0
30
.



 12 CI
 Trang
2

Bài 12: 
60
0
.
1/ 
2/ CMR:
3/T 
xq
S

Bài 13: 



xq
S

Bài 14: 

0
BAA' 45
.
1/ 
2/ 
xq
S


Bài 15: 



2/ C/m :
AN A'B
.
3/ 
4/ 
AMN
S
.
Bài 16: i
A, AB =a,
AC a 3

 

Bài 17: 
AA' a 2
. 

Bài 18: 







Bài 19: 
0
120



. 
xq
S

Bài 20: 

C 




Bài 21: 
A 


Cho BB
xq
S

Bài 22: 

Bài 23: 
0
BAD 60


AB' BD'
. 
 12 CI
 Trang
3
Bài 24: 



.
1/ 
BC

2/ 
Bài 25: EFF

0
60



2)Dạng 2:    có góc    và  

  

0

 
= a ,

ACB
= 60
o
 
0
. 
  

0


 a và
BAD
=60
o


o


* Bài tập tương tự:
Bài 1: 

o
 
3
a2
V
16


Bài 2: 

o
 
3
a3
V
2


Bài 3: 

o
 . 
AB' a 3
;
3
a3
V
2


Bài 4: 
o
ACB 60

o
ch
tam giác ABC'.


3
6Va
, S =
2
3a 3
2

Bài 5: C

0
 
3
32a
V
9


Bài 6: 

o

o



3
a2
V
8



 12 CI
 Trang
4
Bài 7: 
ABCD và OA' = a .T
1) 
2) 
o
.
3) 
o
. 
3
2a 6
V
9

;2)
3
a3
V
4


;3)
3
4a 3
V

9


Bài 8: 


o
.

o
. 
3
a3
16
2)V =
3
a2
8

Bài 9: 

o


3
và S = 6a
2

Bài 10 : ó AB = a ; AD = b ; AA' = c và BD' = AC' =
CA' =

2 2 2
a b c

1) 
2) 

2 2 2
sin x sin y sin z 1  
.


3) Dạng 3:    có góc  2  

 

0

  ABC 

0

 

o
.Tính 
 

o

o




* Bài tập tương tự:
Bài 1:  

o

0


3
2a 2
V
3


Bài 2: 
 
o
 
3


Bài 3: 

o
 
3
V a 2

Bài 4: 
o
BAC 120

o
.
 12 CI
 Trang
5

3
a3
V
8


Bài 5: : 

o
 
3
h2
V
4


Bài 6: . Tính 


1) 
o
.
2) 
o
.
3) 

3
V a 3
; 2) V =
3
a3
4
; V =
3
a3

Bài 7: 

1) 
o
.
2) 
0
.
3)  
3
. 2) V = 12a
3

.3) V =
3
16a
3

Bài 8: . Tính 


o
.


0

3
a6
2
V 
; 2) V =
3
a
; V =
3
a2

Bài 9: 
60
o
. 


o
.

a
2


0

3
3a 3
V
4

; 2) V =
3
3a 2
8
; V =
3
3a
2

Bài 10: .  

1) AB = a :
3
2V 8a
;


o

3
115a
;
3) 30
0
. V =
3
16a




4) Dạng 4:    xiên

 
bên là
a3

o

 



 12 CI
 Trang
6


 
3
AD =
7
.Hai
(AD
0
và 60
0
.




Bài tập tương tự:
Bài 1: 

o
 
3
a2

Bài 2: 

o
 
Bài 3: Cho hình 
o
BAD 30



o
 
abc 3
4

Bài 4 : 

2a 3
3
 
3
a3
V
4


Bài 5:  
 
góc 60
o
.
1) 
2) 
3
3a 3
V
8


Bài 6: 

o

1)  
2)  
2
a3
S
2

2)
3
3a 3
V
8


Bài 7: uông góc

1) 
2)  
o
2)
3
3a
V
8


Bài 8: 

90
o
. :
3
27a
V
42


Bài 9:  
trên mp 
góc 60
o
.
1) BCD.
2) 
3)  
22
ACC'A' BDD'B'
S a 2;S a
. 3)
3
a2
V
2


Bài 10: 
o
. chân

 
o

 12 CI
 Trang
7
 
3
2
3a
V &S a 15
4






: THỂ TÍCH KHỐI CHÓP

1) Dạng 1:  chóp có  bên vuông góc  

 Cho hình chóp SABC có SB = SC và (ASC) cùng

 Cho hình chóp 


o
.


 

o
. 
  vuông góc

o
.
 chóp SABCD.


Bài tập tương tự:
Bài 1: 

o


3
a2
6

Bài 2: Cho hình chóp SABC có SA vuông 

o



3
h3
V
3


Bài 3: 
a, 
o

o

2

= SB
2
+ AB
2
+ AC
2
.  
3
a3
V
27


Bài 4: 

 BC = 5 cm.

1)  
3

2)  
12
34

Bài 5: 
o
BAC 120
,

SA (ABC)
    
o
    

3
a
V
9


Bài 6: 
SA


o

 12 CI

 Trang
8

3
a3
V
48


Bài 7: 

(ABCD), SC

o
và AB = 3a , BC = 4a.  
3

Bài 8: 
o
và
SA

 

3
a2
V
4


Bài 9:   AB=BC=a,
AD=2a, SA


o
. 

3
a6
V
2


Bài 10 :   

o

SABCD. 
3
3R
V
4




2) Dạng 2 :  chóp có   bên vuông góc  

 




 
(ABC)


o
.

 
 45
0
.
a) 
b) .

Bài tập tương tự:
Bài 1: 

1) 
2)  
3
a3
V
24


Bài 2:  
 
góc 45

o
 
3
a
V
12


Bài 3: Cho hình chóp SABC có
oo
BAC 90 ;ABC 30
, 
(SAB)

 
2
a2
V
24


 12 CI
 Trang
9
Bài 4: SH = h và
(SBC)


o



3
4h 3
V
9


Bài 5: 
 
3
a6
V
36


Bài 6 : 


 
3
4h
V
9


Bài 7: 

o

chóp SABCD. 

3
a3
V
4


Bài 8: AB = 2a , BC = 4a,
(SAB)


o

tích hình chóp SABCD. 
3
8a 3
V
9


Bài 9: 


3
a5
V
12


Bài 10: 


chóp SABCD . 
3
a3
V
2




3) Dạng 3 :  chóp 

 Cho SABC  
  
SABC .
Ví dụ 2:


 
a) 


Bài tập tương tự:
Bài 1:  60
o

tích hình chóp. 
3
3a
V
16



 12 CI
 Trang
10
Bài 2: Cho hình chóp tam 
o
.
 
a
3


3
a
V
6


Bài 3:  
o
.
 
3
a3
V
24


Bài 4 :  

o

hình chóp. 
3
h3
V
3


Bài 5 : 
o

tích hình chóp. 
3
h3
V
8


Bài 6 : 
o
ASB 60
.
  
2
a3
S
3


 
3
a2
V
6


Bài 7 : 
o
. Tính

3
2h
V
3


Bài 8: 
o

 a.

3
8a 3
V
3


Bài 9: 
o

.
 
3
a3
V
12


Bài 10: 


3
9a 2
V
2

. 
Bài 11: 
Bài 12:
1/ 


2/ 

. 
Bài 13: 
1/ 
2/ 



Bài 14:

0
30

Bài 15:
1/ 
;
xq tp
SS

 12 CI
 Trang
11
2/ 
3/ 
Bài 16:
R3


0
30
.
1/ 
;
xq tp
SS

2/ 
Bài 17:


1/ 
;
xq tp
SS

2/ 
Bài 18:
1/ 
2/ 
3/ 
Bài 19:
0
60
.
1/ 
2/ 
3/ 
Bài 20:
OM = x (0<x<h).
1/ 
()


()


Bài 21:

.

1/ 

tan

Bài 22: 

1/ 
2/ 
xq
S


tp
S

Bài 23:
ASB
.
1/ 
xq
S

2/ 
2
a
cot 1
22



3/ G



Bài 24:

0
60

Bài 25:, AB=AC=5a ,BC =6a, 

0
60

Bài 26:

AD SB, AE SC
 SA=c.
 12 CI
 Trang
12
1/ 
2/ p(SAB) .
Bài 27:

Bài 28:
ao cho AM =3MD.
1/ 
2/ 
Bài 29:



Bài 30:
ng
0
60

ABCD.
Bài 31:

ABCD
V(H)
V
.
Bài 32:
Bài 33:
Bài 34:
Bài 35:
. CMR:
S.A'B'C'
S.ABC
V
SA ' SB' SC'
. . .
V SA SB SC


Bài 36:

0

60

Bài 37:

0
60

Bài 38:,,
AD=b, SA =c.   SD sao cho
AB' SB,AD' SD
. 

Bài 39:

0
60


Bài 40:
1/ 
2/ 
ABC


Bài 41:

1/ 
2/ 

(H)

(H')
V
V

Bài 42:

ABC
.
 12 CI
 Trang
13

2/ CMR:
SC mp(AB 'C')
.
3/ 
Bài 43:2a,
ABC

0
CAB 30
. 


2/ CMR:
AH SB

SB mp(AHK)
.


Bài 44: 2a ,SA=a ,
SB a 3



Bài 45: 

SB,BC,CD. CMR:
AM BP

Bài 46: 

:
MN BD

Bài 47: ,
0
ABC BAD 90
, BA=BC=a ,AD =2a. 

SA a 2
. 
m
SCD

 
d H;(SCD)
.
Bài 48: 
a. 


Bài 49: 
AD a 2
, 
SA mp(ABCD)
. 
1/ Cmr:
mp(SAC) mp(SMB)

2/ 
Bài 50: 
SA mp(ABC)
. 

Bài 51: 



Bài 52: 






Bài 53: 

. 

Bài 54:




1/ CMR:
2/ 
Bài 55: .


. 
xq
S


 12 CI
 Trang
14
Bài 56: 
, 

. 
0
A 90
,
0
B 60


xq
S


Bài 57: 
A2



. 
xq
S


Bài 58: Ch

ASB
= 2


 
00
0 45  
. 
xq
S

Bài 59: 
0
ASC 90


. 
Bài 60: 

0
BAC 90 ,ABC  
; 
(SAB)
(ABC)
. 
Bài 61:
2


xq
S

Bài 62:

 
d S;(ABC)
.
Bài 63:
a3


AHK
S
.
Bài 64:
2
a3



0
60


0
45
.


Bài 65:, AB=BC= 2a



Bài 66:
1/ C/m:
SA SC

2/ 
Bài 67:

0
45
.


 
d C;(SBD)
.
Bài 68:
0

ABD ABC 60
,
0
CBD 90


Bài 69:
1/ 
2/ 
Bài 70:
26
. 

 12 CI
 Trang
15
Bài 71:

0
BOH COH 30

A sao cho OA =OB .
1/ 
2/ 
 
d O;(ABC)
theo h .
Bài 72:
1/ C/m :
SA SC

.
2/ 
Bài 73:
a3
.
Bài 74:
0
ASB 90
,
0
BSC 60
,
0
ASC 90
.
1/ C/m :
ABC

2/ 
Bài 75:
CH AB


sao cho
0
ASB 90
.
1/ C/m :
SHC


2/ 
Bài 76:

Bài 77:

2. 3IS a
.
1/ C/m:
SAD

2/ 
 
d C;(SAD)
.
Bài 78:


0
45

xq
S

Bài 79:
0
A 120
.

a3
.

1/ 
2/ 
mp(ABC).
Bài 80:

a2
. 
Bài 81:
AC=2a ,AD=3a.
1/ 
 
d A;(BCD)

2/ 
BCD
S
.
Bài 82:
1/ 
2/ 
Bài 83:


(
00
45 90 )  

TP
S


 12 CI
 Trang
16
Bài 84:
SA=
a5

1/ 
2/ 
Bài 85:
SAB 



.
Bài 86:

1/ 
TP
S

2/ 
SB
,
AF SD
. C/m:
SC mp(AEF)
.
Bài 87:
SA=SB =SC= SD =a. 

TP
S

Bài 88: Cho S.ABC 
mp(ABC)


1/ 
 
d A;mp(SBC)
.
2/ 
 
d O;mp(SBC)
.
Bài 89:, AB=AD
=a, 
mp(ABCD)
, SD= a .
1/ C/mr:
SBC

SBC
S
.
2/ 
 
d A;(SBC)
.
Bài 90:


a2

Bài 91:,AB=AD=a,

mp(ABCD)
,SD
a3

SC

(K
SC)

SC mp(EBK)
.
Bài 92:
SA (ABCD)
, SA=
a6


1/ C/m :
AH (SBC)

2/  
 
d O;(SBC)
.
Bài 93:


1/ 
SBD
S
.
2/ 
Bài 94:

a2
. 
xq
S
,
tp
S

Bài 95:





1/ 
2/ 
 
d E;(SAB)
.

 12 CI
 Trang
17
4) Dạng 4 :  chóp &  pháp    tích

 
2AC a
, SA vuông góc

SA a
.




) qua AG và song song 

: 
AB a


CD a


a) .
b) 
()CE ABD

c) .
 

)(



: 
góc
60



a) 
b) 
c) 
: Cho hình c
2SA a
.
a) 
b) 
( ' ')SC AB D

c) 

Bài tập tương tự:
Bài 1: Ch
 
1
k
4


Bài 2: 
3
,trên AB,A

3

Bài 3: 
a 2a
AB ;AC'
23

'C'D . 
3
a2
V
36


Bài 4: 
3

 s: V = 1 m
3

Bài 5: 
a3
 


3

a3
V
40


Bài 6: 
3
 

hình chóp SA'B'C'D'. 
3

 12 CI
 Trang
18
Bài 7: Cho hình chóp SABCD c
3



3

Bài 8: 

 
2
ah
V
9


Bài 9 : 


1
k
2


Bài 10: 
SM
x
SA



51
x
2




5) Dạng 5 : Ôn   chóp và  

 Cho hình chóp S.ABCD có ABCD là hình vu 

60





 Cho hình chóp tam giác S.ABC có AB = 5abên SAB, SBC,
SC
o

: 
3AB a


a) 
b) 
c)  
: 

: 
a) 
b) 

Bài tập tương tự:
Bài 1: 
1
B
1
C
1
có ABC vuông. AB = AC = a; AA
1
= a
2

. M là trung

1

1
BC
1
 V =
12
2
3
a

Bài 2: Hình ch

(ABC).
ACB
= 60
o
, BC = a, SA = a
3
,
  V
MABC
=
3
4
1
a

Bài 3: SABCD có 
ACB
= 90
o
. 

3
 SABCD =
6
4

 12 CI
 Trang
19
Bài 4: 
a) 
o

2
12

b) AB = 1, SA = 2 . 
11
12

Bài 5. a,
AC=a
3
Tính V theo a?


3
a
2

Bài 6: ABCD =
3


o

o
. Tính VSABCD.

3
V
3


Bài 7: Cho hình chóp SABC có SA = SB = SC = a, ASB = 60
o
, BSC = 90
o
, CSA = 120
o
. 
SABC . 
a2
V
12


Bài 8: 
3a
t

Tính t:
3
.
3
3
S BMDN
a
v 

Bài 9: ng a. M,

ra. 
Bài 10: Cho hình chóp S.A
 BC,
CD. 

3
.
3
96
M CNP
a
v 

 12 CI

 Trang
20
Bài tập ôn tập hình không gian



1. 
2. 
3. 
4. 
2
ABC
= S
2
OBC
+S
2
OAC
+S
2
OAB

5. 
6. Ch
7. 
8. 
2
OBC
= S
ABC

.S
HBC

9. 

,

,


cos
2

+cos
2

+cos
2

=1
10. 
2222
1111
OCOBOAOH


11. Cho OA = a, OB = b, OC = c. E 
12. 

AOH=


,

BOH=

,

COH=


sin
2

+sin
2

+sin
2

=2.
13. 

AOM=

,

BOM=

,


COM=


cos
2

+cos
2

+cos
2

=1


Bài 1: 

 
Bài 2: 

2a
,
SA =
3a
.



Bài 3: 
3a

.
Bài 4: Cho hình c

0
60
.


Bài 5: Cho 
sao cho AM = 3MD.


Bài 6: 
sao cho AM = 3MD.


Bài 7*: 
B = a,
SB' 2
SB 3


 12 CI
 Trang
21
Bài 8*: 
a
2

Bài 9*: 

0
ABC 60
, AB = a (a > 0), H là trung

0
30
.
ình chóp S.ABC.

Bài 10*: 
·
BAC = α
,

0
60
.


.





I/. KHỐI CHÓP
Bài 1: Cho 

2





Bài 2: 
SA=AC , AB=a và góc
0
45ABC 

Bài 3 :Cho hình chóp tam giác 
6
.
M,N là trung 
Bài 4: 




Bài 5:
o

hình chóp SABCD theo a
Bài 6: 
Tính eo a.
Bài 7: Cho hình chóp tam giác S.ABC có  ABC là tam giác vuông   B,  bên SA vuông
góc   SA = AB = BC = a. Tính    chóp S.ABC
Bài 8: Cho hình chóp  giácS.ABCD có  ABCD là hình vuông   a ,  bên SAvuông góc
  và SA = AC . Tính    chóp S.ABCD 
Bài 9:
Cho hình chóp S.ABC có áy là tam giác ABC vuông  B, ng th SA vuông góc

i m
phng
(
ABC
)
. 

AB
=
a, BC
=
a3


và SA
=
3a.
(Thi TNTHPT 2008 ln 1)
1. Tính thhi chóp S.ABC theo a.
2. Gi I là trung  cSC, tính  dài n thng BI theo a.
Bài 10:
Cho hình chóp S.ABC có áy là tam giác ABC vuông  B, ng th SA vuông góc

 12 CI
 Trang
22
i mhng
(
ABC
)

. 
AB
=
a, BC
=
a
3


và SA
=
3a.
(Thi TNTHPT 2008 ln 2)

1. Tính thhi chóp S.ABC theo a.
2. Gi I là trung  cSC, tính  dài n thng BI theo a.
Bài 11: 

0

[TNTHPT 2009]
Bài 12: 
góc v. Cho SA = a.

BSC 
và
asin
AB
cos2




.

Bài 13: 



Bài 14: 
ASB 2
.
 :
2
a (1 cot )
)
 :
3
2
a
cot 1
12



 
3
a
12

. 

arccot 2

Bài 15: Cho 

 
3
a3
6

 
15
arctan
5


Bài 16
 : a
3


3
12

Bài 17

0

.

 : V =

a
3

6
24

Bài 18

0

. Tí : a
3


3
8

Bài 19

0

.
a/
 12 CI
 Trang
23
 : V =
a
3

3
3
; AH =
a 3
2

Bài 20

0

 :
a
3

2
6

Bài 21

0


:
h
3

3
3

Bài 22


0


0



2

= SB
2

+ AB
2

+ AC
2

 :
a
3

3
27

Bài 23

 : V = 8 cm
3

. d =
12
34

Bài 24


BAC = 120
0

 SA 

0


: V =
a
3

9

Bài 25 y

0

 : V =
a
3

3
48

Bài 26 

0

 : V = 20a
3


Bài 27
0

và SA 
 : V =
a
3

2
4

Bài 28
AD = 2a, SA   60
0

 : V
=
a
3

6
2

Bài 29

0


: V =
3R
3

4

Bài 30


 : V=
3
1
S
ABCD
. SH =
6
3
3
a

 12 CI

 Trang
24
Bài 31 

(BCD)

o

: V =
3
1
S
BCD
.AH =
2
1
.
3
1
BC.HD.AH =
9
3
3
a

Bài 32: Cho hình chóp S.ABC c

o
.


 : V
SABC
=
3
1
S
ABC
.SH =
12
3
a

Bài 33


b/. Tính . : V =
a
3

3
24

Bài 34

0

. Tính
 : V =
a
3

12

Bài 35: Cho hình chóp SABC có


BAC = 90
0

à (SAB)  
 : V =
a
2

2
24

Bài 36
(SBC)  
0


: V =
4h
3

3
9

Bài 37

 S:
a
3

6
36

Bài 38

a/
 : V =
4h
3

9

Bài 39 

0


 : V =
a
3

3
4

Bài 40: Cho hình chóp SABCD có ABCD là hì (ABCD),


0


SABCD. : V =
8a
3

3
9

 12 CI
 Trang
25
Bài 41

: V =
a
3

5
12

Bài 42

SABCD. : V =
a
3

3
2

Bài 43

: a
3


11
12

Bài 44

 : V =
a
3

2
6

Bài 45


: V =
a
3

2
24

Bài 46

0


chóp. : V =
3a
3

16

Bài 47 
0

.
 
a
3

 :
a
3

6

Bài 48: Cho 
góc 60
0

 : V =
a
3

3
24

Bài 49: Cho hình chóp tam giác 
0


tích hình chóp. : V =
h
3

3
3

Bài 50 
0

. Tính
 : V = h
3


3
8

Bài 51


ASB = 60

0

.
 
a
2

3
3

 
a
3

2
6