12.5 Volumes of Pyramids and Cones Essential Question

12.5 Volumes of Pyramids and Cones
TEXAS ESSENTIAL
KNOWLEDGE AND SKILLS
G.10.B
G.11.D
Essential Question
How can you find the volume of a pyramid
or a cone?
Finding the Volume of a Pyramid
Work with a partner.
The pyramid and the prism
have the same height and
the same square base.
h
When the pyramid is filled with sand and poured into the prism, it takes three
pyramids to fill the prism.
ANALYZING
MATHEMATICAL
RELATIONSHIPS
To be proficient in math,
you need to look closely
to discern a pattern
or structure.
Use this information to write a formula for the volume V of a pyramid.
Finding the Volume of a Cone
Work with a partner. The cone and
the cylinder have the same height
and the same circular base.
h
When the cone is filled with sand and poured into the cylinder, it takes three
cones to fill the cylinder.
Use this information to write a formula for the volume V of a cone.
Communicate Your Answer
3. How can you find the volume of a pyramid or a cone?
Section 12.5
Volumes of Pyramids and Cones
671
What You Will Learn
12.5 Lesson
Find volumes of pyramids.
Find volumes of cones.
Core Vocabul
Vocabulary
larry
Use volumes of pyramids and cones.
Previous
pyramid
cone
composite solid
Finding Volumes of Pyramids
Consider a triangular prism with parallel, congruent bases △JKL and △MNP. You can
divide this triangular prism into three triangular pyramids.
L
P
N
N
M
M
L
L
P
M
N
K
J
K
Triangular
prism
L
J
M
Pyramid Q
M
P
N
J
L
Triangular
pyramid 3
Triangular
pyramid 2
You can combine triangular pyramids 1 and 2 to form a pyramid with a base that is a
parallelogram, as shown at the left. Name this pyramid Q. Similarly, you can combine
triangular pyramids 1 and 3 to form pyramid R with a base that is a parallelogram.
K
N
K
Triangular
pyramid 1
M
K
Pyramid R
L
— divides ▱JKNM into two congruent triangles, so the
In pyramid Q, diagonal KM
bases of triangular pyramids 1 and 2 are congruent. Similarly, you can divide any cross
section parallel to ▱JKNM into two congruent triangles that are the cross sections of
triangular pyramids 1 and 2.
By Cavalieri’s Principle, triangular pyramids 1 and 2 have the same volume. Similarly,
using pyramid R, you can show that triangular pyramids 1 and 3 have the same
volume. By the Transitive Property of Equality, triangular pyramids 2 and 3 have
the same volume.
The volume of each pyramid must be one-third the volume of the prism, or V = —13 Bh.
You can generalize this formula to say that the volume of any pyramid with any base is
equal to —13 the volume of a prism with the same base and height because you can divide
any polygon into triangles and any pyramid into triangular pyramids.
Core Concept
Volume of a Pyramid
The volume V of a pyramid is
h
V = —13 Bh
where B is the area of the base
and h is the height.
h
B
B
Finding the Volume of a Pyramid
Find the volume of the pyramid.
SOLUTION
V = —13 Bh
(⋅ ⋅)
Formula for volume of a pyramid
= —13 —12 4 6 (9)
Substitute.
= 36
Simplify.
The volume is 36 cubic meters.
672
Chapter 12
Surface Area and Volume
9m
6m
4m
Finding Volumes of Cones
Consider a cone with a regular polygon inscribed in the base. The pyramid with the
same vertex as the cone has volume V = —13 Bh. As you increase the number of sides of
the polygon, it approaches the base of the cone and the pyramid approaches the cone.
The volume approaches —13πr 2h as the base area B approaches πr 2.
Core Concept
Volume of a Cone
The volume V of a cone is
V = —13 Bh = —13πr 2h
where B is the area of the base,
h is the height, and r is the
radius of the base.
h
h
B
B
r
r
Finding the Volume of a Cone
Find the volume of the cone.
4.5 cm
2.2 cm
SOLUTION
V = —13 πr 2h
=
1
—3 π
Formula for volume of a cone
⋅ (2.2) ⋅ 4.5
2
Substitute.
= 7.26π
Simplify.
≈ 22.81
Use a calculator.
The volume is 7.26π, or about 22.81 cubic centimeters.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Find the volume of the solid.
1.
2.
5m
20 cm
8m
12 cm
Section 12.5
Volumes of Pyramids and Cones
673
Using Volumes of Pyramids and Cones
Using the Volume of a Pyramid
O
Originally,
Khafre’s Pyramid had a height of about 144 meters and a volume of about
22,218,800 cubic meters. Find the side length of the square base.
SOLUTION
V = —13 Bh
2,218,800 ≈
Formula for volume of a pyramid
1
—3 x2(144)
Substitute.
6,656,400 ≈ 144x2
Multiply each side by 3.
46,225 ≈ x2
Divide each side by 144.
215 ≈ x
Khafre’s Pyramid, Egypt
Find the positive square root.
Originally, the side length of the square base was about 215 meters.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
3. The volume of a square pyramid is 75 cubic meters and the height is 9 meters.
Find the side length of the square base.
4. Find the height of the
5. Find the radius of the cone.
triangular pyramid.
V = 24 m3
13 in.
h
r
3m
V = 351π in.3
6m
Changing Dimensions in a Solid
Describe how multiplying all the linear dimensions by —13 affects the volume of the
rectangular pyramid.
SOLUTION
18 m
Before change
Dimensions ℓ= 9 m, w = 6 m, h = 18 m
9m
6m
Volume
1
V = — Bh
3
1
= —(9)(6)(18)
3
= 324 m3
After change
ℓ= 3 m, w = 2 m, h = 6 m
1
V = —Bh
3
1
= —(3)(2)(6)
3
= 12 m3
1
Multiplying all the linear dimensions by — results in a volume that
3
12
1
1 3
is — = — = — times the original volume.
324 27
3
()
674
Chapter 12
Surface Area and Volume
Changing a Dimension in a Solid
Describe how doubling the height affects the volume
of the cone.
ANALYZING
MATHEMATICAL
RELATIONSHIPS
Notice that when the
height is multiplied by
k, the volume is also
multiplied by k.
1
Voriginal = — π r 2h
3
1
Vnew = — π r 2(kh)
3
1
= (k)— π r 2h
3
= (k)Voriginal
5 ft
3 ft
SOLUTION
Before change
Dimensions r = 3 ft, h = 5 ft
Volume
1
V = — πr 2h
3
1
= —π (3)2(5)
3
= 15π ft3
After change
r = 3 ft, h = 10 ft
1
V = — πr 2h
3
1
= —π (3)2(10)
3
= 30π ft3
30π
Doubling the height results in a volume that is — = 2 times the
15π
original volume.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
6. In Example 4, describe how multiplying all the linear dimensions by 4 affects the
volume of the rectangular pyramid.
1
7. In Example 4, describe how multiplying the height by —2 affects the volume of the
rectangular prism.
8. In Example 5, describe how doubling the radius affects the volume of the cone.
9. In Example 5, describe how tripling all the linear dimensions affects the volume
of the cone.
Finding the Volume of a Composite Solid
Find the volume of the composite solid.
6m
SOLUTION
Volume of
Volume of
Volume of
=
+
solid
cube
pyramid
= s3 + —13 Bh
5 cm
6m
Write formulas.
⋅
= 63 + —13 (6)2 6
Substitute.
= 216 + 72
Simplify.
= 288
6m
6m
Add.
The volume is 288 cubic meters.
10 cm
3 cm
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
10. Find the volume of the composite solid.
Section 12.5
Volumes of Pyramids and Cones
675
12.5 Exercises
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. REASONING A square pyramid and a cube have the same base and height. Compare the volume of
the square pyramid to the volume of the cube.
1
2. COMPLETE THE SENTENCE The volume of a cone with radius r and height h is —3 the volume
of a(n) __________ with radius r and height h.
Monitoring Progress and Modeling with Mathematics
In Exercises 3 and 4, find the volume of the pyramid.
(See Example 1.)
3.
11. Volume = 224 in.3
12. Volume = 198 yd3
4.
7m
3 in.
12 in.
16 m
4 in.
12 m
11 yd
8 in.
9 yd
3 in.
13. ERROR ANALYSIS Describe and correct the error in
finding the volume of the pyramid.
In Exercises 5 and 6, find the volume of the cone.
(See Example 2.)
5.
6.
13 mm
1m
2m
10 mm
✗
V = —13 (6)(5)
5 ft
= —13 (30)
= 10 ft3
6 ft
In Exercises 7–12, find the missing dimension of the
pyramid or cone. (See Example 3.)
7. Volume = 912 ft3
8. Volume = 105 cm3
15 cm
19 ft
14. ERROR ANALYSIS Describe and correct the error in
finding the volume of the cone.
✗
10 m
V = —13 𝛑 (6)2(10)
= 120𝛑 m3
6m
7 cm
s
9. Volume = 24π m3
w
15. OPEN-ENDED Give an example of a pyramid and a
10. Volume = 216π in.3
18 in.
h
r
3m
676
Chapter 12
Surface Area and Volume
prism that have the same base and the same volume.
Explain your reasoning.
16. OPEN-ENDED Give an example of a cone and a
cylinder that have the same base and the same
volume. Explain your reasoning.
In Exercises 17–22, describe how the change affects the
volume of the pyramid or cone. (See Examples 4 and 5.)
18. multiplying all the
17. doubling all the
linear dimensions by
linear dimensions
27.
28.
12 in.
4 in.
12 in.
5.1 m
12 in.
20 cm
5.1 m
29. REASONING A snack stand serves a small order of
7 in.
popcorn in a cone-shaped container and a large order
of popcorn in a cylindrical container. Do not perform
any calculations.
16 cm
12 cm
19. multiplying the base
3 in.
3 in.
20. tripling the radius
edge lengths by —13
8 in.
11 ft
5m
22. multiplying the height
by —32
height by 4
6 in.
30. HOW DO YOU SEE IT?
The cube shown is
formed by three
pyramids, each with
the same square base
and the same height.
How could you use this
to verify the formula
for the volume of
a pyramid?
14 cm
10 in.
12 cm
18 cm
In Exercises 23–28, find the volume of the composite
solid. (See Example 6.)
24.
7 cm
5 cm
In Exercises 31 and 32, find the volume of the right cone.
31.
10 cm
5 cm
9 cm
$2.50
a. How many small containers of popcorn do you
have to buy to equal the amount of popcorn in a
large container? Explain.
b. Which container gives you more popcorn for your
money? Explain.
9 ft
21. multiplying the
8 in.
$1.25
9m
23.
5.1 m
1
—4
32.
22 ft
60°
32°
14 yd
8 cm
12 cm
3 cm
3 cm
25.
33. MODELING WITH MATHEMATICS
26.
8 ft
10 cm
9 ft
12 ft
A cat eats half a cup
of food, twice per day.
Will the automatic
pet feeder hold enough
food for 10 days?
Explain your reasoning.
(1 cup ≈ 14.4 in.3)
2.5 in.
7.5 in.
4 in.
Section 12.5
Volumes of Pyramids and Cones
677
34. MODELING WITH MATHEMATICS During a chemistry
lab, you use a funnel to pour a solvent into a flask.
The radius of the funnel is 5 centimeters and its height
is 10 centimeters. You pour the solvent into the funnel
at a rate of 80 milliliters per second and the solvent
flows out of the funnel at a rate of 65 milliliters
per second. How long will it be before the funnel
overflows? (1 mL = 1 cm3)
39. MAKING AN ARGUMENT In the figure, the two
cylinders are congruent. The combined height of
the two smaller cones equals the height of the larger
cone. Your friend claims that this means the total
volume of the two smaller cones is equal to the
volume of the larger cone. Is your friend correct?
Justify your answer.
35. ANALYZING RELATIONSHIPS A cone has height h
and a base with radius r. You want to change the cone
so its volume is doubled. What is the new height if
you change only the height? What is the new radius
if you change only the radius? Explain.
36. REASONING The figure shown is a cone that has been
warped but whose cross sections still have the same
area as a right cone with equal radius and height. Find
the volume of this solid. Explain your reasoning.
3 cm
2 cm
37. CRITICAL THINKING Find the volume of the regular
40. MODELING WITH MATHEMATICS Nautical deck
prisms were used as a safe way to illuminate decks
on ships. The deck prism shown here is composed of
the following three solids: a regular hexagonal prism
with an edge length of 3.5 inches and a height of
1.5 inches, a regular hexagonal prism with an edge
length of 3.25 inches and a height of
0.25 inch, and a regular
hexagonal pyramid with
an edge length of 3 inches
and a height of 3 inches.
Find the volume of the
deck prism.
pentagonal pyramid. Round your answer to the
nearest hundredth. In the diagram, m∠ABC = 35°.
A
41. CRITICAL THINKING When the given triangle is
C
B
rotated around each of its sides, solids of revolution
are formed. Describe the three solids and find their
volumes. Give your answers in terms of π.
3 ft
38. THOUGHT PROVOKING A frustum of a cone is the
part of the cone that lies between the base and a plane
parallel to the base, as shown. Write a formula for
the volume of the frustum of a cone in terms of a, b,
and h. (Hint: Consider the “missing” top of the cone
and use similar triangles.)
a
20
15
25
42. CRITICAL THINKING A square pyramid is inscribed in
a right cylinder so that the base of the pyramid is on
a base of the cylinder, and the vertex of the pyramid
is on the other base of the cylinder. The cylinder has
a radius of 6 feet and a height of 12 feet. Find the
volume of the pyramid.
h
b
Maintaining Mathematical Proficiency
Reviewing what you learned in previous grades and lessons
Find the indicated measure. (Section 11.2)
43. area of a circle with a radius of 7 feet
44. area of a circle with a diameter of 22 centimeters
45. diameter of a circle with an area of
46. radius of a circle with an area of
529π square inches
256π square meters
678
Chapter 12
Surface Area and Volume