Chapter	6
 Solid	Geometry
 Questions	 about	 solid	 geometry	 frequently	test	 plane	 geometry
 techniques.	 They’re	 diɽcult	 mostly	 because	 the	 added	 third
 dimension	makes	them	harder	to	visualize.	You’re	likely	to	run	into
 three	or	four	solid	geometry	questions	on	either	one	of	the	Math
 Subject	Tests,	however,	so	it’s	important	to	practice.	If	you’re	not
 the	artistic	type	and	have	trouble	drawing	cubes,	cylinders,	and	so
 on,	it’s	worthwhile	to	practice	sketching	the	shapes	in	the	following
 pages.	The	ability	to	make	your	own	drawing	is	often	helpful.
 PRISMS
 Prisms	are	three-dimensional	ɹgures	that	have	two	parallel	bases
 that	 are	 polygons.	 Cubes	 and	rectangular	solids	 are	 examples	of
 prisms	that	ETS	often	asks	about.	In	general,	the	volume	of	a	prism
 is	given	by	the	following	formula:
               Volume	of	a	Prism
                   V	=	Bh
 In	this	formula,	B	represents	the	area	of	either	base	of	the	prism
 (the	top	or	the	bottom),	and	h	represents	the	height	of	the	prism
 (perpendicular	 to	 the	 base).	 The	 formulas	 for	 the	 volume	 of	 a
 rectangular	solid,	a	cube,	and	a	cylinder	all	come	from	this	basic
 formula.
                                 Area and Volume
                              In general the volume of a
                              shape involves the area of
                              the base, often referred to
                              as B, and the height, or h,
                                    of the solid.
 RECTANGULAR SOLID
 A	rectangular	solid	is	simply	a	box;	ETS	also	sometimes	calls	it	a
 rectangular	 prism.	 It	 has	 three	 distinct	 dimensions:	length,	 width,
 and	height.	The	volume	of	a	rectangular	solid	(the	amount	of	space
 it	contains)	is	given	by	this	formula:
         Volume	of	a	Rectangular	Solid
              V	=	lwh
 The	surface	area	(SA)	of	a	rectangular	solid	is	the	sum	of	the	areas
 of	all	of	its	faces.	A	rectangular	solid’s	surface	area	is	given	by	the
 formula	on	the	next	page.
       Surface	Area	of	a	Rectangular	Solid
           SA	=	2lw	+	2wh	+	2lh
 The	volume	and	surface	area	of	a	solid	make	up	the	most	basic
 information	you	can	have	about	that	solid	(volume	is	tested	more
 often	 than	 surface	 area).	 You	 may	 also	 be	 asked	 about	lengths
 within	a	rectangular	solid—edges	and	diagonals.	The	dimensions	of
 the	solid	give	the	lengths	of	its	edges,	and	the	diagonal	of	any	face
 of	a	rectangular	solid	can	be	found	using	the	Pythagorean	theorem.
 There’s	 one	 more	 length	 you	 may	 be	 asked	 about—the	 long
 diagonal	 (or	space	 diagonal)	 that	 passes	 from	 corner	 to	 corner
 through	the	center	of	the	box.	The	length	of	the	long	diagonal	is
 given	by	this	formula:
   Long	Diagonal	of	a	Rectangular	Solid	(Super	Pythagorean
                       Theorem)
                     2   2   2    2
                    a 	+	b 	+	c 	=	d
 This	is	the	Pythagorean	theorem	with	a	third	dimension	added,	and
 it	 works	 just	 the	 same	 way.	 This	 formula	 will	 work	 in	 any
 rectangular	box.	The	long	diagonal	is	the	longest	straight	line	that
 can	be	drawn	inside	any	rectangular	solid.
 CUBES