':... Chapter 6 Solid Geometry
(luestions abol,t solid geornctrv fLeqrenrly resr ptanc geomerry rcchniques. They're diflicuir mosdy bccause rl.e 3ddeo rhird di,nen.ron m-ke" .lr
Volume of a Prism
Area and Volume In this fornula, B rcpresents rhc area of eirher base of the prism (rhc r<,p or the ln lenrra ihe vclume of a bottom), and I represcnrs rhe height ofthe prisnr (perperdicular ro rhe base). lhc shirle frolves the area oi lormulas lor che volume of a rccrangular solid, a cube, and a cylinder all come the base, oltef referred t. from this basic formula. as B, afd lre hel!hi. o. r. of ihe so id
RECTANGULAR SOLID
A recrangular solid is siDDly a bor; ETS also somctimes calls it a rcctengular prism. Ir has rlrrcc distinct dimensions: /rzgrl, ridth, and height.The $\une of a rectangul:r solirl (the anlount ofspace ir conrains) is given by this formula:
Volume ofa Rectangular Solid
v = /r,h
Th€ surface arca fS,4) ofa rectangular $lid is rhe sum ofthe .rreas ofall of its faces. A rectangular solid\ surface area is given by the formula on the next page.
130 Crark ng the SAT N/lath 1& 2 SublectTests b,- *s
/
Surface Area ofa Rectangular Solid SA-2lw+2uh+)lh ne he The volurne and surface area ofa solid make*rp the mosr basic information vou can \ave rbouL rhrr .olid ,vol,,re i5 re,'ed more ofren r\an u-rr.e area,. you mr1 al'o b. arkeo b.1gh.w\h.nd rccLangutar .otrd- "bour edge, .rnd diagon"ls. The dtmen.ior. o'rhe.olid g;ve .he iengrh, or ir. edgc,. and rLe di"gon,l oFrnyl.r ofa rcctanguiar solid can be lound using the pyrt rgo.""" tn.oi.-. T"..;. o". more length you may be asked aboLit rhe long diagonal (or space diagonal) that f:on come.r tu corner through the lisses center of e bo". Th; lensth;fthe l""r d;agonal is given by this formula:
Llar
Long Diagonal ofa Rectan$lar Solid (Super pythagorean 'Iheorem)
Ihis is rhe Pythagorean theorem with a rhird dimension added, and it works just *re same way. This formula will work in any recrangdar box. Th€ long diatonJ- is rle longest straight liae thar can be drawn inside any rectangular solij.
So rd Geometrv I l3t CUBES
diarensions All A cube is a rectangular solid rhat has the same lengfi in all rhree s; of its face. areiq"ares 'lliis simplifies the fonnuhs for volume' sLrrface area' and the long d;agonal.
Volurne ofa Cube
surface Area ofa Cube
Face Diagonalofa Cube
rong Diagonal of a Cube
132 Cracking lhe SAT lv4ath 1 & 2 SublectTesls l \:3
/ - -tt
CYTINDERS
A cylinder is iike a prism but with a circular base. It has two irnportanr dimen- sions-radius and heighr Remember that volume is the area ofrhe base tirnes the height. In this case, the base is a circle. The area ofa circl€ is nl. So the volume of t cyllnder is nl h. E}
Volume of a Cylinda V=rlb
Solid Geometry 133 'Ihe surface area of a cylinder is iound by:rdding the rreas of the two circular bases ro the arel ofthe rectangle you'd get ifyou unrolled the side ofthe cylinder' That boils down to rhc following fornula: c G
Surface Area of a Cylinder SA=2nr'+2rh
The longest line that can be drawn inside a cylinder is rhe djagonal ofthe rectan gle forrned by d,e diameter and the height ofthe cylinder' You can 6nd its lengdr wirh rhe Pythagorean theorem.
r,- -+:
2t
,t' = Q,)'* h'
134 I Cracking the SAT t"lath I &2 SubrectTesls GONES
f,you take a cylinder and shrink one ofim cirular bases down to a poinr, then n have a cone. A cone has three significanr dime'sions which form a righr tri ade-its radius, irs hei ght, and i* tunt height, svhich is the straightJine distance 6m the tip of the cone ro a point on the edge of its base. The lornulas for the ntrme and surface aea ofa cone lrre given in rhe information box at rhe begin- ing of both of rhe Math Subiecr Tests. The fornula lor the volurne of a cone is gntty straightforward:
Connectths Dob Notice that the volume of a cofe is iust one'third of the vohme ola circ! ar cylifder. fulake memorlzing eas\/l you have ro be careful comprdrrg surface area for a cone using the formuta 'Ihe by ETS. fornula ac rhe beginning ofthe Math Subject Tests is lor Iateral area of z cor.e-the area of the sloping sides-not rhe complete surface It doesnt indude rhe circular base. Heret a more useful equarion for the
solid Geometry | 135 Surface Area ofa Cone
Ifyou want to calculate only the lateral area ofa cone, jusr use rhe 6rst halfofthe above forrnula-learc the zl o11.
136 CraDk ng the SAT N4ath I & 2 Subjectlests SPHERES
A sphere is simply a hollow ball. k can be defined as all of rhe points in space at a 6xe ...--t.. Volume ofa Sphere 3 Surface Arca of a Sphere sA = 4nt frre- Ee intersecrion ofa plane and a sphere always forms a circle unless the plane is r-?,t to the sphere, in which case the plane and sphere touch at onty one point. Solid ceometry | 137 PYRAMIDS A pyrxmid is 1 lide lil Connect the Dots N!lac lhai Tr. ra L,:,r! .i r n!iam i is l.r!1r rc'iuril ol Lh!,/r. |fre o{r I lsr \la[e iefior iin!]easy Volume ofa Pyrarnid v =L nt 3 E-draofr,$e) Tt\ nor Leallv posible ro give a gcncLal formuh for the surface area of r pyr:rmid because there are so rrany differenr kinds. At anv rate, rhe informatn,n is not gen erally tcsrcd on the Math Subiecr Tests. Ifyou should be cailed ttpon to figurc our rhc suLfacc area of .r pyramid, just 6gure out rhe area of each face using polygon rules, rnd add them up. 138 Crack nu the SAT tu1ath 1 & 2 SubjertTesLs TRICKS OF THE TRADE Here are some of the most common solid geometry question types youie likely to encounter on the Marh Subject Tests. They occur much more often on the Math levei 2 Subject Test than on the Math Level 1 Subject Tesr, but they can appear Triangles in Rectangular Solids Many questions about rectangular solids are actually testing triangte rules. Such questions generally ask for the lengths of the diagonals of a boxt faces, the long diagonai of a box, or other lengrhs. These questions are usually solved using rhe Pythagorean rheorem and rhe Super Pychagorean thebrern that iinds a boxt long diagonal (see the section on RecrangLrlar Solids). DRITL Here are sor"e praciice questions using triangle rries in rectangular,solids. The ^,} znswers to these drills can be found in Chapter I 2. l\ 32. What is the lenglh of the longest line that can be q \ drawn in a cube of volume 27 ? {l (A) 3.0 (B) 4.2 br (c) 4.9 + (D) s.2 \0 (E) 9.0 -t.. \l$ D H 7 G F B 36. In the rectangular solid shown, if, B = 4, BC = 3, and Af - I 2. \^har i\ lhe perirnerer of triangle EDB 1 (A) 27.33 (B) 28.40 (ct 29.20 (D) 29.s0 (E) 30.37 solld Georn€try | 13e ' 39- In the cube above. M is the midpoint ofBC' and N i\ ttre mrdournl ol L,H Il lhe crrbe h:r' r \olume of L whrt is ttie length ol MN I (A) 1.23 (B) 1.36 (c) l.4l (D) L73 (E) 1.89 Volume Ouestions rl'' reL'ior '\rp,h'' Vany .oliJ gco-ler-' quFrion' reJ vo'rlrnoe 'r"ndrng o .o'rd. lr' orhcr J:r' n':orr'- omeri-('rntlud'-Brr'e lests 1r0 I Crackrng the SAT Nrlalh 1 & 2 Subjed )rill :ie ftillowing pracrtce questions. Thc ansri-$s to rhcse drills c.rn bc found in Chaprer 12. lhe volume and surlace area of a cube are equal \\'hat is the ]ength ol an edge of lhis cube? 1) I Bl l C) ,1 Dr 6 43 A sphere ofradius I is tohlly submerged in il - icctangul solid has a volume of 30. and its cylinddcal tank ofradius:1- as sho\rn. The lrrater .:sc\ have integer lcngLhs. \Vhat is the greatest levei in the tnnk rises a distance ofh. What is thc r tiible surlace rrea ol this solicll value of h ? -61 :81 (A) 0.072 -N6 (B) 0.083 1 9.1 (c) 0.096 :l]l (D) 0.108 (E) 0.r23 --: ,rarcr in Allegra's swimming pool has a .. of 7 f-eet. If the area of rhe pentagonal base -: oool is 150 square fcer, then what is the n. A cubc has a surface irrea of6r. What is thc vol, -,r- in cubic leet. of the water in her pool? umc of the cuhe! l (A) (B) j] (c) 6x' (D) (E) :16. A sphere has a ndius of r. lfthis radius is in- creased by ,. then the surfilce area of the sphere is increased by what amount l (A) &' (B):ltt&l (C) ihrl) +,{nrl (D) 8r||b +2rb + b2 (.E) 4rFbz So d G€ometr\, I lat 40. If the pyramld shown has a square base with edges ot'length D. and, = 2r, then which ofthe following is the volume of the pyramid? rAr 4 (B) 3 (c) 4l?' (D) 8n' l' (E) 3 142 Cracking the SAT Vlalh 1 & 2 SubjectTests lnscribed Solids Some quesrions on the Math Subject Tests wili be based on spheres inscribed in cube. or . ube' in Following are a few basic tips that can speed up your work on inscribed solids $fsrions. Vhen a cube or recangular solid is inscribed in a sphere, the long diagonal of rhat solid is equai to the diameteldfthe sphere. \Vhen a cylinder is ;nscribed in a sphere, the sphere's diamerer is equal to the diagonal of the recangle formed by the cylinder's heights Sohd Geornetry l{3 \(hen a sphere is inscribed in a cube, the diameter of the sphere is equal to the length of dre cubet edge. Ifa sphere is inscribed in a cylinder, borh solids have the same Most inscribed solids questions fali into one ofthe preceding categories lfyou run irro a siruation not covered by these rips, just look for rhe w.ry ro get from rhe dimensions oflhc iDner shape to rhose of thc exrernal shape, or vice versa. DRlLL Here ar€ some practice inscribed solids quesrions The answem to these drilLs can be found in Chrpter 12. 32. A rectangular solid is inscribed in a sphere as shown. lf the dimensions of the solid are 3. 4. and 6. then what is thc radius ofthe sphere? (A) 2.49 (B) 3.91 (c) 4.r6 (D) 5.62 (E) 7.81 144 I Crack ns the SAI Math 1 & 2 SubjertTens 35. A cylinder is inscribed in a cube with an edge of Iength 2. What volume of space is erclosed by the cube but not by the cylinder? (A) 1.41 (B) 1.56 (c) r.72 (D) 3.81 (E) 4.86 38. A cone is inscribed in a cube of lolume I in such a way that its base is inscribed in one face of the cube. What is the volume of the cone? (A) 0.21 (B) 0.26 (c) 0.33 (D) 0.42 (E) 0.67 lrlids Produced by Rotation types ofsolids can be produced by the rotation of simple rwo-dimensional es-sphera, cylinders, and cirnes. Qtestions about solids produced by rota- are generally fairly simple; they usually test your abtlity to visualize the solid ated by the roration ofa far shape. Somerimes, roered solids qu€stions bedn a shape in the coordinate plan€-rhat is, rocared around one ofrhe axeror other iine. Practice will help you 6gure out the dimensions of the solid from dirnensions ofrhe originat flar shape. Soid Gsom€ty ] 145 rorJlei J are'er' fli' r' an er'' A :. produ.' d *he- i-'lc I 'nrrrd 't' 'p\erc ' ' IL e wirr \rle rl-t .,, 1,,a. ," rir\. a' l-e 'phcre r 'o r\( o-iginrl 'rme ".,f out anvdringvou *ent to .#;:'il ;;;';rthe.ircLe, and vou ca" ngurc around a cenrral line ar one is forrned bY rhe rotrrion of a rectangle edge. of its legs (think ofit as by rotating a right triangie around onc A cone is lormed ;r' ol ,.,''""*. orh' ' o*el" Lr;r rgle rro:nd "\i' ')m r'l'-'"* '''"ng '" rri:nere in rhc ,.r ;;;:;" hi',r.rn5 rbou rt;' ir'"u 'ourr r\c ' ,|.,',, i";' rn3 on il::: ,i.*;;;i ,*1 ,-i" 'l' '"'"'g ''"'"a 'he'es ' 'r'ling '**rg"re l:ke"s^e :flou'pun rhe rhi d 5gu'e I'r'". r* -o' e' 'e'"nd 'h' r r a* uF { -m"rr)r' vo'[r :ii:i.".;;,T"-^t.l "-"ri'a .n 1"u,- ,o',. ng 'ro'nd would end up rvith'; thc fourth ligure Lests 146 I C'ackinq the SAT Math I & 2 SLrblect DBIIL Try these rotated solids questions for practice. The answers to these drilts can be forind in Chapter 12 34. What is the volume of rhe solid generated by rctating rectangle ABCD around AD ? .(A) 15.? (B) 31.4 (c) 62.8 (D) ',72.0 (E) .80.0 39. If the triangle created by OAB is rotated around the i-axis, what is the volume of the gene.ated solid? (A) 15.70 (B) 33.33 (c) 40.00 (D) 4'7.t2 (E) 78.54 sorid Geomeq I r47 46. What is the volume generated by rotating square ABCD around the -r-axis? (A) 24.84 (B) 28.27 (c) 42.66 (D) 56.s5 (E) 84.82 Changing Dimensions so*. iliJg.o'"",,y qLrestions will ask vou to figure out what happens to the of sol;d ifall ofits lengths are increased bv a cenain frctor or if irs area "ol"me " dorbles, and so on. To answer questions ofthis rvpe, just remember a basic rule' Vhen rhe lengths of a solnl are increased bv a cerrain factor, the surface the increases by rhe square of rhat factor, and rhe volurne in- area "f -lid .re:.e. bi rhe cub< o' rl'J tu.ror. Ihit rule i' rrue on', rl'en rFc 'olid ' .h"pe d"e'n\ , \,nge :r.lengrhmu't in. rei\c ir r'"n J imen ru'] "or run. ."d ofren used lor th;s tvpe of or,.'. r- ,h.. r."'oi', ct,I,., 'ph-.' "'" -o" question because rhen shapes are consrant. 148 I Crackins the SAI [r]alh 1& 2 SubjeciTests i1 E: : .aulltl d, nf)i.J) D.zt , tLzt .jrc .o..c,pez.//bF 13. Ifthe radius of sphercA is one,third as long as the radius of sphere B, then the volume of sphcre ,4 is what fiaction ofthe volume oSe ofsphere B ? I ule. (A) 3 (B) I l (c) q I (D) 12 l (E) )'7 lli. A rcctangular solid wi$ length l. widrh ]r, and height , has a volume of 24- What is the volume tn , la r(crdngrlir.!,liLl'n sithlcn!th' 2-.u Jth- )' and height : ? (A) l8 (B) r2 (c) 6 (D) 3 (E) 2 So d Geomer'y I l4s I In the illustration above, a length is doubled, which means that the corresponding afta is 4 dmes as $eat, ard the volume is 8 times as great. Ifthe length had been cipled, the area would have increased by a factor-of 9, and the volume by a factor af 27. DRILL Try these practice questions. The answers to these drills can be lound ir Chaprer 12. 13. If the radius of sphere A is one third as long as the radius of sphere B, then the volume of sphere , A is what fraction of tbe volume of sphere B ? l (A) ; I (B) ; I (c) t I (D) D (E) I -)1 18. A rcctangular solid with length l, widrh )r, and height l, has a volume of 24. What is the volume Iw of a rectangular solid with length -. width , h ; and height t? (A) 18 (B) 12 (c) 6 (D) 3 (.E) 2 Solid Geometry ] l{9 21. Ifthe surface aler o[ a cube is incieased by a fac tor of2.25, fhen its volume is increased by what (A) r.72 (B) 3.38 (c) 1.so (D) s.06 (E) s.64 150 I Crackiflq tlre SAT [4ath 1 &2 SubjectTesls Summary Solid geometLy quesdons are often plane gcometr) quesrions in disguise. For rhc purposes ofrhe SAT Math I & 2 Sub, jecr lesls, prisms are 3,dimensional figures with mo parallel, identical bases. The Lrases can be anv shape lrom plane geometry The volLrme of a prism is rhe area ol rhe ba*. often refcned to as B, rimes rhe heighr, l. Lers talk recrangular prisms: . lle forrnLrl.r for rhe volLrme ofa rectangu- lar prlsn is V = /uh. 'the . lormula 6r rhe surfic dea of x recr rngular solid is SA - 2/a + 2wh + 2/h. Think about paintjng rhc outside of rhe ligure. Find the area ofeach side. . The Super Pythagorean theorem, which is hclpfLrl ln solving questions about thc dlagonal ofa recranguhr prim, is i+ b1 +c1 -d]. Let's ralk cubes. Remember thar a cube is jusr a rectangular prism whose lengrh, widih, and height are equa1. Ifyorr forger a formula, jusr use the rcctangular prisrr formulal . The vohme ofa cube is r= i. ' lL".urla.. .rcr oi" re.r"ngu"r.o.J s,4 = 6;. Let's talk cylinders. A cylinder is a prisrn rhosc .'Ilevolumeof a ci4i^det is V = Ttlh. . The surfice arca of a recrangular solid i. (/ 'rl + )n.h. , yoLr lorgcr Ir.. remember $at youie jusr painring c ou ''de \o ),,u ll nerd rhe .rrc' of r,.u .r.1.. ,.'d rhe a ." ". r\e o h. p,e.e. rvhich, rhen rolled our (like a ro11 ofpaper towels), is a rectangle rvhose sides are the . ircuml"rc-, c o' rhe, .lc r d rhe h. ighL l 151 A cone is similar ro a cvlinder excepr thar one ofits bascs is meiclv a point. . ll'. or ur. l" rl-e ro'u e oi., .o . : l Y- t /, qh r. rl - h.i,hr n,-'r L. ] perpendicular ro rhe brse. . '11,e fomula for rhc surfacc arca of a con. is Jll - n,l + nrr. q,heLe /is rhe siant hcigfr. A ryhcrc is a hollov ball. . l\eror .rl lor..inc^ .r't. j v =1nJ 3 . 11,e forrnLrla lonhe surf.rce .rrea oi a conc is .17 = 4rr:. l'r'rrmnLs rre like cones, Lrut the b.rse is .r planc gcomcrrv sh,rpc. l1,c formula for rhl'olLLnrc of rD\LrDriJi\ L =-Br. ', 3 In . .l h3 1... or .. n. rhat connccts dit inner figurc to the outcr iigure. Qucsiions rboui rnids produccd b.v roration usurlh tcst vouL rbilitv to visualize the solnl createcl bi rhe rorarion ol.r llat dupe. 152 Cracking Lhc SAI lr'larlr 1 & 2 Subl€ctTests