# The Tich Khoi Chop Va Khoi Lang Tru

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Chuyn :Luyn tp Hnh Hc Khng Gian

GV: L Minh Tin

CHUYN :

PHNG PHP LUYN TP TH TCH KHI A DIN

N TP 3 KIN THC C BN HNH HC LP 12 A. TH TCH KHI A DIN I/ Cc cng thc th tch ca khi a din: 1. TH TCH KHI LNG

TR:V= B.h vi B:d t h ie c a n y h a u h: c ie c oc ba

h B

Th tch khi hp ch nht: V = a.b.c vi a,b,c l ba kch thc Th tch khi lp phng: V = a3 vi a l di cnh

aa

a

2. TH TCH KHI CHP:V= Bh1 3h

B : die tch a n y vi u h: chie cao 3. T S TH TCH T DIN: Cho khi t din SABC v A, B, C l cc im ty ln lt thuc SA, SB, SC ta c:VSABC VSA'B'C' S S S A B C = S S A' B'S C'A '

B

S C' A'

A

B' C B

B ' C '

4. TH TCH KHI CHP CT:V= h B+ B' + 3

A

B

(

BB'

)

C

B, B': die tch hai a n y vi u h: chie cao II/ Bi tp:

Chuyn :Luyn tp Hnh Hc Khng Gian

GV: L Minh Tin

LOI 1:

TH TCH LNG TR

1) Dng 1:

Khi lng tr ng c chiu cao hay cnh y

V d 1: y ca lng tr ng tam gic ABC.ABC l tam gic ABC vung cn ti A c cnh BC = a 2 v bit A'B = 3a. Tnh th tch khi lng tr.A' B' 3a A a B C C'

a 2

Li gii: Ta c VABC vung cn ti A nn AB = AC = a ABC A'B'C' l lng tr ng AA ' AB VAA 'B AA '2 = A 'B2 AB2 = 8a2 AA ' = 2a 2 Vy V = B.h = SABC .AA' = a3 2

V d 2: Cho lng tr t gic u ABCD.ABCD' c cnh bn bng 4a v ng cho 5a. Tnh th tch khi lng tr ny. C' Li gii: D' ABCD A'B'C'D' l lng tr ng nn A' BD2 = BD'2 - DD'2 = 9a2 BD = 3a B' 3a 4a 5a ABCD l hnh vung AB = 2 C D 2 9a Suy ra B = SABCD = A B 4 Vy V = B.h = SABCD.AA' = 9a3 V d 3: y ca lng tr ng tam gic ABC.ABC l tam gic u cnh a = 4 v bit din tch tam gic ABC bng 8. Tnh th tch khi lng tr.A' B' C'

Li gii: Gi I l trung im BC .Ta c V ABC u nn AI = A 'I BC(dl3 ) 2S 1 SA'BC = BC.A 'I A 'I = A'BC = 4 2 BC AA ' (ABC) AA ' AI . VA 'AI AA ' = A 'I2 AI2 = 2 Vy : VABC.ABC = SABC .AA'= 8 3AB 3 = 2 3 & AI BC 2

A I B

C

V d 4: Mt tm ba hnh vung c cnh 44 cm, ngi ta ct b i mi gc

Chuyn :Luyn tp Hnh Hc Khng Gian

GV: L Minh Tin

tm ba mt hnh vung cnh 12 cm ri gp li thnh mt ci hp ch nht khng c np. Tnh th tch ci hp ny.D' C'

D'A' D

C' C

D'B' C

D

C'

A'A B

AA'

BB'

B'

Gii Theo bi, ta c AA' = BB' = CC' = DD' = 12 cm nn ABCD l hnh vung c AB = 44 cm - 24 cm = 20 cm v chiu cao hp h = 12 cm Vy th tch hp l V = SABCD.h = 4800cm3

V d 5: Cho hnh hp ng c y l hnh thoi cnh a v c gc nhn bng 600 ng cho ln ca y bng ng cho nh ca lng tr. Tnh th tch hnh hp .C' B' D B C

D'

Li gii: Ta c tam gic ABD u nn : BD = aa2 3 v SABCD = 2SABD = 2

A'

Theo bi BD' = AC = 2

A

60

a 3 =a 3 2 VDD'B DD' = BD'2 BD2 = a 2 a3 6 Vy V = SABCD.DD' = 2

2)Dng 2:

Lng tr ng c gc gia ng thng v mt phng.

V d 1: Cho lng tr ng tam gic ABC A'B'C' c y ABC l tam gic vung cn ti B vi BA = BC = a ,bit A'B hp vi y ABC mt gc 600 . Tnh th tch lng tr.A' C'

B'

A 60o B

C

Li gii: Ta c A 'A (ABC) A 'A AB& AB l hnh chiu ca A'B trn y ABC . ABA Vy gc[A 'B,(ABC)] = ' = 60o VABA ' AA ' = AB.tan 600 = a 3 1 a2 SABC = BA.BC = 2 2 a3 3 Vy V = SABC.AA' = 2

V d 2: Cho lng tr ng tam gic ABC A'B'C' c y ABC l tam gic

Chuyn :Luyn tp Hnh Hc Khng Gian

GV: L Minh Tin

vung ti A vi AC = a , ACB = 60 o bit BC' hp vi (AA'C'C) mt gc 300. Tnh AC' v th tch lng tr. o A' C' Li gii: VABC AB = AC.tan60 = a 3 . Ta c: AB AC;AB AA ' AB (AA 'C'C) nn AC' l hnh chiu ca BC' trn (AA'C'C). B' o 30 Vy gc[BC';(AA"C"C)] = BC'A = 30o AB VAC'B AC' = = 3a tan30o C A a V =B.h = SABC.AA' o 60 VAA 'C' AA ' = AC'2 A 'C'2 = 2a 2 B 2 VABC l na tam gic u nn SABC = a 3 2 3 Vy V = a 6

V d 3: Cho lng tr ng ABCD A'B'C'D' c y ABCD l hnh vung cnh a v ng cho BD' ca lng tr hp vi y ABCD mt gc 300. Tnh th tch v tng din tch ca cc mt bn ca lng tr . Gii: A' D' Ta c ABCD A'B'C'D' l lng tr ng nn ta c: DD' (ABCD) DD' BD v BD l hnh chiu ca BD' trn ABCD . o C B Vy gc [BD';(ABCD)] = DBD' = 300 30 D a 6 A VBDD' DD' = BD.tan 300 = 3 a 3 a 6 4a 2 6 Vy V = SABCD.DD' = S = 4SADD'A' = 3 3 V d 4: Cho hnh hp ng ABCD A'B'C'D' c y ABCD l hnh thoi cnh o o a v BAD = 60 bit AB' hp vi y (ABCD) mt gc 30 . Tnh th tch ca hnh hp.C' B'B' C'

Gii VABD u cnh a SABD =C

A'

D' B

a2 3 4

o 30 A 60 o

a

D

a2 3 2 VABB' vung tiB BB' = ABtan30o = a 3 3a3 V = B.h = SABCD .BB' = Vy 2 SABCD = 2SABD =

3) Dng 3:

Lng tr ng c gc gia 2 mt phng

Chuyn :Luyn tp Hnh Hc Khng Gian

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A'

A

V d 1: Cho lng tr ng tam gic ABC A'B'C' c y ABC l tam gic vung cn ti B vi BA = BC = a ,bit (A'BC) hp vi y (ABC) mt gc 600 .Tnh th tch lng tr. C' Li gii: Ta c A 'A (ABC)& BC AB BC A 'B B' ABA Vy gc[(A 'BC),(ABC)] = ' = 60o VABA ' AA ' = AB.tan 600 = a 3 C 1 a2 o 60 SABC = BA.BC = 2 2 B a3 3 Vy V = SABC.AA' = 2

V d 2: y ca lng tr ng tam gic ABC.ABC l tam gic u . Mt (ABC) to vi y mt gc 300 v din tch tam gic ABC bng 8. Tnh th tch khi lng tr. Gii: VABC u AI BC m AA' (ABC) C' A' nn A'I BC (l 3 ). Vy gc[(A'BC);)ABC)] = A 'IA = 30oB'

A

30o B xI

2x 3 = x 3 .Ta c 2 2 AI 2 x 3 A' AI : A' I = AI : cos 30 0 = = = 2x 3 3

Gi s BI = x AI =

C

3 =x 3 Vy VABC.ABC = CI.AI.AA = x3 3 M SABC = BI.AI = x.2x = 8 x = 2 Do VABC.ABC = 8 3AA = AI.tan 300 = x 3.

V d 3: Cho lng tr t gic u ABCD A'B'C'D' c cnh y a v mt phng (BDC') hp vi y (ABCD) mt gc 60o.Tnh th tch khi hp ch nht.

Chuyn :Luyn tp Hnh Hc Khng GianC' B' D' A'

GV: L Minh Tin

C

60 0

D O Ba

A

Gi O l tm ca ABCD . Ta c ABCD l hnh vung nn OC BD CC' (ABCD) nn OC' BD (l 3 ). Vy gc[(BDC');(ABCD)] = COC' = 60o Ta c V = B.h = SABCD.CC' ABCD l hnh vung nn SABCD = a2 VOCC' vung nn CC' = OC.tan60o = a 6 2 3 a 6 Vy V = 2

V d 4: Cho hnh hp ch nht ABCD A'B'C'D' c AA' = 2a ; mt phng (A'BC) hp vi y (ABCD) mt gc 60o v A'C hp vi y (ABCD) mt gc 30o .Tnh th tch khi hp ch nht. Ta c AA' (ABCD) AC l hnh chiu ca A'C trn (ABCD) . Vy gc[A'C,(ABCD)] = A 'CA = 30o BC AB BC A'B (l 3 ) . Vy gc[(A'BC),(ABCD)] = A 'BA = 60o VA 'AC AC = AA'.cot30o = 2a 3 2a 3 VA 'AB AB = AA'.cot60o = 3 4a 6 VABC BC = AC2 AB2 = 3 3 16a 2 Vy V = AB.BC.AA' = 3

A' B' 2a A o 30 C C'

D'

D

o 60 B

4) Dng 4:

Khi lng tr xin

V d 1: Cho lng tr xin tam gic ABC A'B'C' c y ABC l tam gic u cnh a , bit cnh bn l a 3 v hp vi y ABC mt gc 60o . Tnh th tch lng tr.A' B' C'

A a B

C

6 0

o

H

Li gii: Ta c C'H (ABC) CH l hnh chiu ca CC' trn (ABC) C'CH = 60o Vy gc[CC',(ABC)] = 3a VCHC' C'H = CC'.sin 600 = 2 2 a 3 3a 3 3 SABC = = .Vy V = SABC.C'H = 4 8

Chuyn :Luyn tp Hnh Hc Khng Gian

GV: L Minh Tin

V d 2: Cho lng tr xin tam gic ABC A'B'C' c y ABC l tam gic u cnh a . Hnh chiu ca A' xung (ABC) l tm O ng trn ngoi tip tam gic ABC bit AA' hp vi y ABC mt gc 60 . 1) Chng minh rng BB'C'C l hnh ch nht. 2) Tnh th tch lng tr . Li gii: 1) Ta c A 'O (ABC) OA l hnh chiu ca AA' trn (ABC) Vy gc[AA ',(ABC)] = OAA ' = 60o B' Ta c BB'CC' l hnh bnh hnh ( v mt bn ca lng tr) AO BC ti trung im H ca BC nn o BC A 'H (l 3 ) 60 A BC (AA 'H) BC AA ' m AA'//BB' C O nn BC BB' .Vy BB'CC' l hnh ch nht. a H 2 2a 3 a 3 2) VABC u nn AO = AH = = 3 3 2 3 B VAOA ' A 'O = AO t an60o = a a3 3 Vy V = SABC.A'O = 4 V d 3: Cho hnh hp ABCD.ABCD c y l hnh ch nht vi AB = 3 AD = 7 .Hai mt bn (ABBA) v (ADDA) ln lt to vi y nhng gc 450 v 600. . Tnh th tch khi hp nu bit cnh bn bng 1.A' C'D' C'

A' B'

Li gii: K AH ( ABCD ) ,HM AB , HN AD A' M AB, A' N AD (l 3 ) A 'MH = 45o,A 'NH = 60o

D N A C H M

t AH = x . Khi 2x AN = x : sin 600 = 33 4x 2 AN = AA' A' N = = HM 3 M HM = x.cot 450 = x2 2

B

3 4x 2 3 Ngha l x = x= 3 7 Vy VABCD.ABCD = AB.AD.x 3 = 3. 7. =3 7

Chuyn :Luyn tp Hnh Hc Khng Gian

GV: L Minh Tin

LOI 2: 1) Dng 1:

TH TCH KHI CHP

Khi chp c cnh bn vung gc vi y

V d 1: Cho hnh chp SABC c SB = SC = BC = CA = a . Hai mt (ABC) v (ASC) cng vung gc vi (SBC). Tnh th tch hnh chp . Li gii: Ta c (ABC) (SBC) AC (SBC) (ASC) (SBC) 1 1 a2 3 a3 3 Do V = SSBC .AC = a= 3 3 4 12

A

a_ B \ S

C /

/

V d 2: Cho hnh chp SABC c y ABC l tam gic vung cn ti B vi AC = a bit SA vung gc vi y ABC v SB hp vi y mt gc 60o. 1) Chng minh cc mt bn l tam gic vung . 2)Tnh th tch hnh chp . Li gii: 1) SA (ABC) SA AB &SA AC m BC AB BC SB ( l 3 ). Vy cc mt bn chp l tam gic vung. 2) Ta c SA (ABC) AB l hnh chiu ca SB trn (ABC). Vy gc[SB,(ABC)] = = 60o . SAB a VABC vung cn nn BA = BC = 2 2 1 a SABC = BA.BC = 2 4 a 6 VSAB SA = AB.tan60o = 2 2 1 1a a 6 a3 6 Vy V = SABC .SA = = 3 34 2 24

S

A 60o

a

C

B

V d 3: Cho hnh chp SABC c y ABC l tam gic u cnh a bit SA vung gc vi y ABC v (SBC) hp vi y (ABC) mt gc 60o. Tnh th tch hnh chp .

Chuyn :Luyn tp Hnh Hc Khng GianS

GV: L Minh Tin

A 60 o a B M

C

Li gii: Ml trung im ca BC,v tam gic ABC u nn AM BC SA BC (l3 ) . Vy gc[(SBC);(ABC)] = = 60o . SMA 1 1 Ta c V = B.h = SABC .SA 3 3 3a VSAM SA = AM tan60o = 2 1 1 a3 3 Vy V = B.h = SABC .SA = 3 3 8

V d 4: Cho hnh chp SABCD c y ABCD l hnh vung c cnh a v SA vung gc y ABCD v mt bn (SCD) hp vi y mt gc 60o. 1) Tnh th tch hnh chp SABCD. 2) Tnh khong cch t A n mt phng (SCD). Li gii: 1)Ta c SA (ABC) v CD AD CD SD ( l 3 ).(1) Vy gc[(SCD),(ABCD)] = SDA = 60o . VSAD vung nn SA = AD.tan60o = a 3 1 1 a3 3 Vy V = SABCD .SA = a2a 3 = 3 3 3 2) Ta dng AH SD ,v CD (SAD) (do (1) ) nn CD AH AH (SCD) Vy AH l khong cch t A n (SCD). 1 1