':... 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<rn r rrder ru vM alize. V,u're likely to mn into three or lour solid geon- etry quesrions on eirhcr .,ne olrhc Mattr Subjcct Tests, howevea so itls imporL.urr to practicc. Ifyouie nor dre artistic type and have trouble drawing cubes, cylindcrs, jct and so on, worthwhile to pracricc skerching rhc shapes in the following pages. llc abiliry to make your own drawing is often helpfut. PRISMS Prisms ire rhrec-diDrensional figures thar have rwo parallel bases rhar arc poly- gons. Cubes and recrangular solids are examples of prisms that ETS ofien asks aboLrt. ln gcneral, d,e volLlme ofa prism is gjven br rhe follos-iDs formulal 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<l disance fiom a central poinr. The one irnponillt measure in r sphereis its ra- 'Ihe dius. fonnulas for the volume and surhce area ofa sphere are given to you ar the very beginning of both Math Subject Tests. Thar means rhar you dont need ro have drern memorized, but here they are anJ.way: ...--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<e a cone, except rhat irs basc is a polygon instead ofa circlc. Pyramids don\ show up ofiur on $e lr4arh Subject rcsts. When rou do run into a pyrarnld, ft will almosr always have a rectangular base. Pyramids can be preny complicated solids, bur f<,r rhe purposes ofrhe Madr Subjecr Tcsts, a pyramid has jusi nio impo.rant nreasLrres-dre area of its basc and irs height. The height ofa pvramid i rhe length of a line dLarvn srraighr dorvn fiom rhe pyrarnld's tip to its b.rse. Thc heighr is perpcndicular ro rhe b:rse.'Ihe volume ot a pyranid is given by rhis fornuh. Connect the Dots N!lac lhai Tr. ra L,:,r! .i r n!iam i is l.r!1r rc'iuril ol Lh!,/r. |fre o{r I lsr \la[e iefior iin!]easy Volume ofa Pyrarnid v =L nt 3 E-draofr,$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.