11-~ Volume of Pyramids and Cones

The volume of a pyramid is one third the product of the area of the base and the height of the pyramid.

1V = -Bh3

Because of Cavalieri's Principle, the volume formula is true for all pyramids, including oblique pyramids. Theheight of an oblique pyramid is the length of the perpendicular segment from the vertex to the plane of thebase.

The volume of a cone is one third the product of the area of the base and the height of the cone.

V = ~Bh, or V = ~"r2.1c This formula applies to all cones, Jluding oblique cones.

Examples: Find the volume of each of the following solids.

1. I b:: \.\C\/1 to~

\j~ 4Q(\O)

~

d \ ~~ -.: '!O ~ "3 C;M7cm

3.

,22 in.

5.

~~ ~(\fS)CL-L.,)

~ \C\$3

,,=- \ C\ ~ ( \ ~)3

~-:. ~ 5 <is' .-Vtv

2.B=-8\1I

\j -:.~Ilf (fl.)------

'b -:.q 'T\

Ci\ = C1'\\ (1)7 rn :. ~3'\\ '" ~

GoV\e:. '11\ (4: : \~\\ro~~

r':-!' /W SI":N 3So\\6. ~ It> ~ \\ + \2 \\ ~ '1 \\N'

4.

"l.. '1-' 2-1?> -=- \ 'l -:.\4'-\ -C--\ ~ 1. -t -{ "2.. -:.- ~

'?(\SM~ \l)~(\~) i1..-:.~\'(-3Co-:'~U?

12 ft '; \ '1:t~ ~ ~ 't '!.- ~ 1112ft P"\V' - ~u, l~-r1): 9("fttlr~

'b

So\leA - ~I:l 'l? 4- q (" il) \-\-j\2

7. Find the radius of a circular cone whose volume is

B~: an~~ght;s 6'1" \""- a';""

i ; 'i\:~~I~ - S- \"2...

2'\\ -: 1\ Y' • lo3

2-~41\ ~ (o't\('

4 -:..'('''2-

9. Find the volume of a circular cone whose radius is12 ft and whose surface area is 300TT ft2.

SA:. LA 'r B I300)1-= ~(g,4)f)~.\. \4-41

3Db ~ I ';)..Q ~\ \ l.ftt

\5~ -=- \ ~ .9. I _ \ L\l\ \\ (5)\ ~ -=- i 'J... ---;-

I) ,J= :A40'lr~~ j10. A 1200 sector is cut from a circle with a radius of 9 in. The 1200 sector is "r~lled" to create a cone.Find t~~;i:?~.*-~~.o~f.the resulting cone. ?J'Z."'" -R 1. ':. q~

,';lllilillil1jrMi, cw: .&n"tj - ~ (\~1I) ~,'"g, I-q ~1 2

lllli~lltlIJf_~~ ~~o ~ ~ q~ (~~) ~if{2d C1

~:;. 3(81\i'):JI; L.f\:: irk o;\d..i ~~~~0Jte-CL. '\ -:. .3 ~ c ~ 11

I y--11. The plane region is revolved completely about the given line to sweep out a solid of revolution. Describethe solid and then ~ind its volume in terms of TT. . \ y(a) about the x-cxrs (b) about the y-axIs 1 1

~ wi;:;tfu ~ CMt cnctl

I"I/.·~ _ .....•

'(":- \.lc ~:= 0 I" ';1/ X

~ = llol\(?) '" '+~~ " ,~ ~ llc1\C~)-=- \61T1

?:> 1.3tSD\\~ ~ L\8 ~ - \\.o'\l :: ~d.ttr u

6.2 ft

6 ft

~--qrrr\/ -=- 9 '1f (2,) ~ k>'\t f\ ~

3

8. Find x if the volume is 126 cm2.

~:: ~ (a.')L'1-) ::

\]-- ~~

I 'd-lP -=- ~.5 'X- I. I ~)

~

318 -=- lo ~ 'X-

Co t-VV) =. ~

Cffi'tL c :.~ ~:. ~

'J ~ ~ rrr (:J3

V~ \d-'"KJ

1

\)