12.5
Volume of Pyramids and Cones
p Find volumes of pyramids and cones.
Goal
Your Notes
THEOREM 12.9: VOLUME OF A PYRAMID
The volume V of a pyramid is
V5
h
,
where B is the area of the base and
h is the height.
Example 1
B
Find the volume of a solid
Describe the solid. Find its volume.
Solution
The solid is a hexagonal pyramid.
10 in.
First, find the area of the base using the
formula for the area of a regular polygon,
4 in.
1
2
} aP. The apothem a of the hexagon is
inches and the perimeter P is
5
inches.
6p
So, the area of the base B is
In an oblique
pyramid the vertex
is not directly over
the center of the
base. By Cavalieri's
Principle, the
volume formula for
a pyramid works for
oblique pyramids.
1
}(
2
)(
)5
4 in.
2 3 in.
in.2.
Then, find the volume.
1
V5}
Bh
3
1
(
5}
3
Formula for volume of a pyramid
)(
) Substitute.
5
Simplify.
<
Use a calculator.
The volume of the pyramid is about
inches.
Copyright © Holt McDougal. All rights reserved.
cubic
Lesson 12.5 • Geometry Notetaking Guide
341
12.5
Volume of Pyramids and Cones
p Find volumes of pyramids and cones.
Goal
Your Notes
THEOREM 12.9: VOLUME OF A PYRAMID
The volume V of a pyramid is
h
1
V5 }
Bh ,
3
where B is the area of the base and
h is the height.
B
Find the volume of a solid
Example 1
Describe the solid. Find its volume.
Solution
The solid is a hexagonal pyramid.
10 in.
First, find the area of the base using the
formula for the area of a regular polygon,
1
2
4 in.
} aP. The apothem a of the hexagon is
}
2Ï3 inches and the perimeter P is
6 p 4 5 24 inches.
So, the area of the base B is
In an oblique
pyramid the vertex
is not directly over
the center of the
base. By Cavalieri's
Principle, the
volume formula for
a pyramid works for
oblique pyramids.
}
}
1
} ( 2Ï 3 )( 24 ) 5 24Ï 3 in.2.
2
4 in.
2 3 in.
Then, find the volume.
1
V5}
Bh
3
Formula for volume of a pyramid
}
1
( 24Ï3 )( 10 ) Substitute.
5}
3
}
5 80Ï3
Simplify.
< 138.56
Use a calculator.
The volume of the pyramid is about 138.56 cubic
inches.
Copyright © Holt McDougal. All rights reserved.
Lesson 12.5 • Geometry Notetaking Guide
341
Your Notes
THEOREM 12.10: VOLUME OF A CONE
The volume V of a cone is
V5
5
,
h
where B is area of the base, h is the
height, and r is the radius of the base.
B
r
Find volumes of cones
Example 2
Find the volume of the cone.
a. Right cone
b. Oblique cone
16.8 mm
12 ft
9.5 mm
4.5 ft
Solution
a. Use the formula for volume of a cone.
1
V5}
πr 2h
3
1
π(
5}
3
)2 (
5
In an oblique
cone the vertex is
not directly over
the center of the
base. By Cavalieri's
Principle, the
volume formula for
a right cone works
for oblique cones.
Formula for volume of a cone
)
π
Substitute.
Simplify.
<
Use a calculator.
The volume of the right cone is about
mm3.
b. Use the formula for volume of a cone.
1
πr 2h
V5}
3
1
π(
5}
3
5
<
Formula for volume of a cone
)2 (
π
)
Substitute.
Simplify.
Use a calculator.
The volume of the oblique cone is about
342 Lesson 12.5 • Geometry Notetaking Guide
ft3.
Copyright © Holt McDougal. All rights reserved.
Your Notes
THEOREM 12.10: VOLUME OF A CONE
The volume V of a cone is
1
1
V 5 } Bh 5 } πr 2h ,
3
h
3
where B is area of the base, h is the
height, and r is the radius of the base.
B
r
Find volumes of cones
Example 2
Find the volume of the cone.
a. Right cone
b. Oblique cone
16.8 mm
12 ft
9.5 mm
4.5 ft
Solution
a. Use the formula for volume of a cone.
1
V5}
πr 2h
3
In an oblique
cone the vertex is
not directly over
the center of the
base. By Cavalieri's
Principle, the
volume formula for
a right cone works
for oblique cones.
Formula for volume of a cone
1
π( 9.5 )2 ( 16.8 )
5}
3
Substitute.
5 505.4 π
Simplify.
< 1587.8
Use a calculator.
The volume of the right cone is about 1587.8 mm3.
b. Use the formula for volume of a cone.
1
πr 2h
V5}
3
Formula for volume of a cone
1
π ( 4.5 )2 ( 12 )
5}
3
Substitute.
5 81 π
Simplify.
< 254.5
Use a calculator.
The volume of the oblique cone is about 254.5 ft3.
342 Lesson 12.5 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Use trigonometry to find the volume of a cone
Example 3
Find the volume of the
right cone.
12 m
12 m
628
628
r
Solution
To find the radius r of the base, use trigonometry.
opp.
628 5 }
adj.
Write ratio.
tan 628 5
Substitute.
r5
<
Solve for r.
Use the formula for the volume of a cone.
1
1
(πr 2) h < }3 π (
V5}
3
)<
)2 (
The volume of the cone is about
cubic meters.
Find volume of a composite solid
Example 4
Find the volume of the solid shown.
The cone and the cylinder are right.
3 cm
7 cm
Solution
Volume of 5 Volume of
solid
1 2
πr h
5 πr 2h 1 }
3
5 π(
5
)2 (
π1
3 cm
1 Volume of
1
)1}
π(
3
π
5
) Substitute.
Simplify.
Use a
calculator.
The volume of the solid is about
centimeters.
Copyright © Holt McDougal. All rights reserved.
)2 (
cubic
Lesson 12.5 • Geometry Notetaking Guide
343
Your Notes
Use trigonometry to find the volume of a cone
Example 3
Find the volume of the
right cone.
12 m
12 m
628
628
r
Solution
To find the radius r of the base, use trigonometry.
opp.
tan 628 5 }
adj.
Write ratio.
12
tan 628 5 }
Substitute.
r
r5
12
< 6.38
Solve for r.
tan 628
Use the formula for the volume of a cone.
1
1
2) h < }
(
V5}
πr
π ( 6.38 )2 ( 12 ) < 511.5
3
3
The volume of the cone is about 511.5 cubic meters.
Example 4
Find volume of a composite solid
Find the volume of the solid shown.
The cone and the cylinder are right.
3 cm
7 cm
Solution
3 cm
Volume of 5 Volume of cylinder 1 Volume of cone
solid
1 2
πr h
5 πr 2h 1 }
3
1
5 π ( 3 )2 ( 7 ) 1 }
π ( 3 )2 ( 3 ) Substitute.
3
5 63 π 1 9 π
Simplify.
5 226.19
Use a
calculator.
The volume of the solid is about 226.19 cubic
centimeters.
Copyright © Holt McDougal. All rights reserved.
Lesson 12.5 • Geometry Notetaking Guide
343
Your Notes
Checkpoint Find the volume of the solid. Round your
answer to two decimal places.
1. Pyramid with regular base
15 cm
9 cm
2. Right cone
10 ft
6 ft
3. Right cone
328
14 yd
4. Composite solid
5.1 m
Homework
5.1 m
5.1 m
344 Lesson 12.5 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Find the volume of the solid. Round your
answer to two decimal places.
1. Pyramid with regular base
15 cm
9 cm
1052.22 cm3
2. Right cone
10 ft
6 ft
301.59 ft3
3. Right cone
328
14 yd
574.82 yd3
4. Composite solid
5.1 m
Homework
5.1 m
5.1 m
97.92 m3
344 Lesson 12.5 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.