1. 12-3 Volume of Prisms &12-3 Volume of Prisms &
CylindersCylinders
p. 743p. 743
NCSCOS: 1.02, 2.03, 2.04NCSCOS: 1.02, 2.03, 2.04

2. Volume of a solidVolume of a solid
• DefnDefn – the amount of space inside the– the amount of space inside the
solid.solid.
• Just think “how much water would itJust think “how much water would it
take to fill the object”?take to fill the object”?
• Measured in cubic units (i.e. cmMeasured in cubic units (i.e. cm33
))

3. Post. 27Post. 27 – Volume of a cube– Volume of a cube
V = sV = s33
V – volume & s – edge lengthV – volume & s – edge length
ExEx: find the volume of the cube.: find the volume of the cube.
V = sV = s33
V = 5V = 533
V = 125 mV = 125 m33
5 m5 m

4. Post. 28Post. 28 – Volume– Volume ≅≅ post.post.
• If 2 polyhedra areIf 2 polyhedra are ≅≅, then they have, then they have
the same volume.the same volume.

5. Post. 29Post. 29 – Volume + post.– Volume + post.
• The volume of a solid is the sum of theThe volume of a solid is the sum of the
volumes of its non-overlapping parts.volumes of its non-overlapping parts.
TotalTotal volumevolume volumevolume
volume =volume = ofof + of+ of
of tower pyramid prismof tower pyramid prism

6. Thm 12.6Thm 12.6 – Cavalieri’s Principle– Cavalieri’s Principle
• If 2 solids have the same height & theIf 2 solids have the same height & the
same cross-sectional area at everysame cross-sectional area at every
level, then they have the same volume.level, then they have the same volume.
• Basically, this means that right &Basically, this means that right &
oblique solids use the same volumeoblique solids use the same volume
formulas, unlike surface area & lateralformulas, unlike surface area & lateral
area formulas.area formulas.

7. Oblique Prisms & Cylinders
• Cavalieri’s Principle:
A right prism and an oblique
prism with the same base
and height have the same
volume.
Cavalieri’s principle also relates to
cylinders. The two stacks have the
same number of CDs, so they have
the same volume.

8. EXAMPLES:
Find the volume of the oblique
cylinder (exact value & nearest
tenth).
V = 5324π ≈ 16,725.8 cm3
Find the volume of the prism.
Round to the nearest tenth if
necessary.
V = 203.7 ft3

9. Thm 12.7Thm 12.7 – Volume of a Prism– Volume of a Prism
V = BhV = Bh
V – volume, B – area of the base, &V – volume, B – area of the base, &
h – height of the prism.h – height of the prism.
ExEx: find the volume of the prism.: find the volume of the prism.
V = BhV = Bh
B = ½ bh = ½ (2.5)(6) = 7.5B = ½ bh = ½ (2.5)(6) = 7.5
V = (7.5)(3)V = (7.5)(3)
V = 22.5 mmV = 22.5 mm33
6mm6mm
2.5 mm2.5 mm 3
m
m
3
m
m

10. Thm 12.8Thm 12.8 – Volume of a Cylinder– Volume of a Cylinder
V = BhV = Bh
OrOr
V =V = ππrr22
hh
ExEx: Find the vol. of the cylinder.: Find the vol. of the cylinder.
V =V = ππrr22
hh
V =V = ππ(4(422
)(13))(13)
V =V = ππ(16)(13)(16)(13)
V = 208V = 208ππ ftft33
or 653.45 ftor 653.45 ft33
13 ft13 ft
4ft4ft

11. ExEx: If the volume of the prism is 396 cm: If the volume of the prism is 396 cm33
,,
then what is the value of x?then what is the value of x?
V = BhV = Bh
396 = (11x)(x)396 = (11x)(x)
396 = 11x396 = 11x22
36 = x36 = x22
6 = x6 = x
x = 6 cmx = 6 cm
11 cm11 cm
xcmxcm
x
cm
x
cm

12. ExEx: Find the: Find the
volume of thevolume of the
concrete block.concrete block.
(whole block) V = Bh = (13.5*4)(3) = 162 in(whole block) V = Bh = (13.5*4)(3) = 162 in22
(1 hole) V = Bh = (3*6)(3) = 54 in(1 hole) V = Bh = (3*6)(3) = 54 in33
(2 holes) V = 2(54) = 108 in(2 holes) V = 2(54) = 108 in33
Volume of concreteVolume of concrete = 162 – 108 = 54 in= 162 – 108 = 54 in33
* If concrete weighs 0.084 lbs/in* If concrete weighs 0.084 lbs/in33
, how much, how much
does this block weigh?does this block weigh?
0.084 * 54 = 4.536 lbs0.084 * 54 = 4.536 lbs
6 in6 in
3 in3 in
13.5 in13.5 in
3in3in
4
in
4
in
HintHint: Find the volume of: Find the volume of
the whole block 1the whole block 1stst
, then, then
subtract out the volume ofsubtract out the volume of
the holes!the holes!