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Tài liệu ôn tập Toán 12

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h
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
 THỂ TÍCH KHỐI ĐA DI Ệ N
I/ Các công thức thể tích của khối đa diện:
1. THỂ TÍCH KHỐI LĂNG TRỤ:


B: dieän tích ñaùy
h : chieàu cao





 Thể tích khối hộp chữ nhật

 !
 Thể tích khối lập phương

"

#$%&



2. THỂ TÍCH KHỐI CHÓP

1
3


B : dieän tích ñaùy
h : chieàu cao



3. TỈ SỐ THỂ TÍCH TỨ DIỆN
'(%)*+++
,#-./012!3$**
*4

SABC
SA ' B' C'
V
SA SB SC
V SA' SB' SC'
=
C'
B'
A'
C
B
A
S

Chú ý:
56!789':;8&%


6!789':<==!>8&%
"

6!789':$=?<4" !%
  
a b c
+ +

56!78':.8,#@&
"

a
"54=#@4=4#,0#8,#@,&A#@B8
C'D4#,0#8,#@:#EF/8G.:#,0
H5IJ8FK#@J8FK#84#,0#8,#@
BÀI TẬP
''=*4#,0.8,;8&#!78L8*;884.=
C
"a
*"
 M - (4=*N'
 OPF8#-.:&* #$%#'&PN'
'4=.8,#@*4&#,0B8&AB8OPF8#-.:

 8.*;884
 M - (4=*PN'

"'4=*4#,0.8,;8&&A*;884#,0
*M - (4=*


H'4=*Q4#,0Q;8&&A*;884#,0&
A*B8
"a

 M - :(4=*Q
 8.F8#-.:&*#-.,#@,#E:4=*Q
R'4=*4*;884S8#;.$*

"a
M - :(4=*
T'(4=*4.D*.8,#@B.F'8.=;884
 - :(4=*
U'(4=*4#,0.8 ;8G&;884:*AC
F/8F8G.O:.8,*3=#,084
V
TV
α
=
M - :(
4=*
W'(4=*Q4#,0Q'*.8,#@&
**QM - :(4=*Q
X'(4=*Q4#,0Q?<'*;884.D#,0CQ
*"
M - :(4=*
V'(4=*Q4#,0Q8;8Y'*;884.D

#,0CQ*Q
M - :(4=*Q
''=*Q4#,0;8&*;884.D#,0CQ84
8?*#,0CQHR
V
M - :(4=*Q
'(4=*4#,0.8,;8Y6E*,#@.D
AC*3=.D#,0C84TV
V
M - (4=*
Q

"'(J8FK+++4#,0.8,#@&B8&AB8
"a
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1) Mặt nón:
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* d: đường sinh
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l: độ dài đường sinh
r: bán kính đường tròn đáy.
3) Khối nón:
4/8=2F'8:4
#!38(4
pThể tích khối nón: 
π
"

F


h: độ dài đường cao
r: bán kính đường tròn đáy
II) MẶT TRỤ, HÌNH TRỤ, KHỐI TRỤ:
1) Mặt trụ:
'#!78L8∆%Z'8Z'8
,.$'q8B8F
\DFoe'0ZY#!78L8%
g0g∆8.DFK
* d: đường sinh
*

: trục
2) Hình trụ:
FKFoe'0ZFY.$
?<g0g.$&
pDiện tích xung quanh:*

eg

π
Fl
l: độ dài đường sinh
r: bán kính đường tròn đáy.
3) Khối trụ:
FK/8=2F'8:4
#!38(FK
pThể tích khối nón: F


h: độ dài đường cao
r: bán kính đường tròn đáy
 Chú ý: #((FKl.
III) MẶT CẦU, HÌNH CẦU, KHỐI CẦU:
1) Mặt cầu:
'#-.j(#lZ(rF
M<=3=,#-.\F'8;88
,#-.j.$'q8B8F#!3
8.D2G.j, F
s )*CjF
{ }
Fj\\
=
Chú ý: pjtF

B.8'C*
pjnF


B.F'8C*
pjF

B.FAC*
2) Vị trí tương đối của mặt phẳng và mặt cầu:
'.D2*CjF.D=L8CuO:jFA.=Cu%j'q8,
Sj#.=Cu
p%tF

Cu;8mC*0Cu

C*
φ
p%F

Cu=evC*&
s#4(S): tiếp diện, (H): tiếp điểm
p%nF

CumC*N'#!78FoC4G., 

%F

 Chú ý:%V0j≡CumC*N'#!78FoCjF
3) Vị trí tương đối của đường thẳng và mặt cầu:
R

'.D2*CjF#!78L8∆O:jFA∆%j'q8,S
j#∆
p%tF


∆;8mC*0∆

C*
φ
p%F

∆=evC*&
s#4∆: tiếp tuyến(H): tiếp điểm
p%nF

CumC*&#-.=G)
HQ) e8g2- (2
pQ) e8g2*
eg
H
π
F


pM- (2
"
H
π
F
"

BÀI TẬP
'4Foe'04#!78'V., #,0FR.
M %) e8g:4#^'

M - :(4
\$%)#g#E:44'q8,SG.:#,0#.D=L8%)
.M %) %)#4
\$FK4, #,0FR.4'q8,8?#,0B8U.
M %) e8g:FK- :(FK
m(FKY.D=L8Z'8Z'84FK,FK".M %) :%)#!3
&'A
"m4B8.$.D=L8gFK:4#!3.$%).$.8,#@&M 
%) e8g- :4#4
H\$FK4, #,0F@'F
"

M %) e8g%) '=2:FK
M - :(FK
'#-.2!3B.FA#!78Fo#,0Z''848?FK:FK
B8"V
V
M 'q8,8?FK:FK
Rm4#E*Y.D=L8#gFK#!3.$.8,;8G4&0@B8


M %) e8g%) #,0- (4
'%G08:#!78Fo#,04Z''.D=L8C*&'.D=L8
#,04.$84TV
V
M %) .8,*
T\D=L8#gFK:FKmFKN'%);8&c
M %) e8g%) '=2:FK
M - :(FK
M - (J8FK8,#@$=FK

U\$(4484Y#EB8V
V
4, #,0B8FM %) :%)#g
#!78Z;884
W\$(J8FK#84@'4#,0.$.8,#@&M - :(FK
8'&=(J8FK0
X\$(%)#@4&B8$=F'8.$(4M - :(4#4
V\$(FK8$=F'8.$(2#!78Fo#,0:(FKB.FA.D:(
2
M %) e8g- :(FK$=F'8.$(2, c
#!78':(FK
M 8,Flw:- (FK$=F'8(2, c'F!
'<==!>8Q+++Q+&
M %) e8g- ::(FK4#!78Fo:#,08'&=,
;8Q+++Q+
M %) e8g- :(44#EG.j:;8Q#,0
#!78Fo$=;8+++Q+
'<==!>8Q+++Q+&k,#lG., .D2#gW
#E:<==!>8#^'
"'%)Q4Q⊥CQR.8,;8&"H
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k,#lG., .D2#g(#E:%)
H'4=.8,#@*4&#,0B8&AB8
k,#lG., .D2#g,#E4=
R '%)Q4Q⊥CQH.8,;8&T
Wk,#lG., .D2#g(#EQ:%)
T'4=*Q4#,0Q;8&*⊥CQ*
k,#lG., .D2#gR#E*Q
U'4=8,#@*Q4&#,0B8&AB8k,#lG.

, .D2#gR#E*Q
W)'J8FK+++4wq,&#@B8
 k,#lG., .D2#g,#E:J8FK
 M %) .D2- :(2!>88
X '4=*Q4#,0;8&*⊥CQQr8.=Cug
;884*\D=L8Cum***Q&++Q+
 \cU#-.Q+++Q+;B.FA.$.D2
 M %) .D2- :(2#!3&'
V '4=.8,#@*4&#,0B8.DA3=#,0.$84B8TV
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PHƯƠNG PHÁP TỌA ĐỘ TRONG KHÔNG GIAN
PHỆ TỌA ĐỘ TRONG KHÔNG GIAN
A. KIẾN THỨC CẦN NHỚ
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zzyyxxAB
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++++
++
==
vv
zyxzyx
zzyyxx
vu
vu
vu
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