Hmh alg1 mod7 2&3 vol pyrds-cones-spheres-2016

Volume of Pyramids and ConesVolume of Pyramids, Cones & Spheres
Warm UpWarm Up
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
Lesson QuizzesLesson Quizzes

Warm Up
1. Find the volume of a rectangular prism that is 4 in.
tall, 16 in. wide, and 48 in deep.
2. A cylinder has a height of 4.2 m and a diameter of
0.6 m. To the nearest tenth of a cubic meter, what
is the volume of the cylinder? Use 3.14 for π.
3. A triangular prism’s base is an equilateral triangle.
The sides of the equilateral triangle are 4 ft, and the
height of the prism is 8 ft. To the nearest cubic foot,
what is the volume of the prism?
3072 in3
1.2 m3
55.4 ft3

Warm Up - continued
4. Find the surface area of a square
pyramid whose base is 3 m on a side
and whose slant height is 5 m.
5. Find the surface area of a cone whose
base has a radius of 10 in. and whose
slant height is 14 in. Use 3.14 for π.
39 m2
753.6 in2

Problem of the Day #1
A ream of paper (500 sheets) forms a
rectangular prism 11 in. by 8.5 in. by 2 in.
What is the volume of one sheet of paper?
0.374 in3

Problem of the Day #2
Find the slant height of the cone with
the following measurements:
The area of its base is one-third of its
total surface area. The radius is 4 cm.
8 cm

Learn to:
•find the volume of pyramids, cones and
spheres.

Vocabulary
pyramid
cone
sphere
hemisphere
great circle

Additional Example 1A: Finding the Volume of Pyramids and
Cones
Find the volume of the figure. Use 3.14 for π.
1
3
V = • 14 • 6
V = 28 cm3
V = Bh
1
3
B = (4 • 7) = 14 cm2
1
2

Additional Example 1B: Finding the Volume of Pyramids and
Cones
1
3
V = • 9π • 10
V = 30π ≈ 94.2 in3
V = Bh
1
3
B = π(32
) = 9π in2
Use 3.14 for π.

Check It Out: Example 1A
1
3
V = • 17.5 • 7
V ≈ 40.8 in3
V = Bh
1
3
B = (5 • 7) = 17.5 in2
1
2
5 in.
7 in.
7 in.

1
3
V = • 9π • 7
V = 21π ≈ 65.9 m3
V = Bh
1
3
B = π(32
) = 9π m2
Use 3.14 for π.
Check It Out: Example 1B
7 m
3 m

Additional Example 2: Exploring the Effects of Changing
Dimensions
A cone has a radius of 3 ft. and a height of 4 ft.
Explain whether tripling the height would have
the same effect on the volume of the cone as
tripling the radius.
When the height of the cone is tripled, the volume is
tripled. When the radius is tripled, the volume becomes
9 times the original volume.

Check It Out: Example 2
A cone has a radius of 2 m and a height of 5 m.
Explain whether doubling the height would have
the same effect on the volume of the cone as
doubling the radius.
Double the
Radius
Double the
Height
Original
Dimensions
1
3
V = πr2
h
1
3
1
3
1
3
= π(22
)5
≈ 20.93 m3
1
3
V = πr2
(2h)
= π(22
)(2•5) = π(2 • 2)2
(5)
V = π (2r)2
h
≈ 41.87 m3
≈ 83.73 m3
1
3
When the height of a cone is doubled, the volume is
doubled. When the radius is doubled, the volume is 4
times the original volume.

Additional Example 3: Social Studies Application
The Pyramid of Kukulcán in Mexico is a square
pyramid. Its height is 24 m and its base has 55
m sides. Find the volume of the pyramid.
B = 552
= 3025 m2
1
3
V = (3025)(24)
V = 24,200 m3
A = bh
V = Bh
1
3
A lowercase b is used to represent the
length of the base of a two-dimensional
figure. A capital B is used to represent
the area of the base of a solid figure.
Caution!

B = 482
= 2304 m2
1
3
V = (2304)(12)
V = 9216 m3
A = bh
V = Bh
1
3
Find the volume of a pyramid with a height
of 12 m and a base with 48 m sides.

Additional Example 4: Using a Calculator to Find Volume
Use a calculator to find the volume of a cone
to the nearest cubic centimeter if the radius
of the base is 15 cm and the height is 64 cm.
Use the pi button on your calculator to find the area of
the base.
2ND ^ × X2 ENTER
Next, with the area of the base still displayed, find the
volume of the cone.
π
15
× 64 × ( )1 3÷ ENTER
The volume of the cone is approximately 15,080 cm3
.
B = πr2
V = Bh
1
3

Use a calculator to find the volume of a cone
to the nearest cubic centimeter if the radius
of the base is 14 cm and the height is 16 cm.
Use the pi button on your calculator to find the area of
the base.
2ND ^ × X2 ENTER
Next, with the area of the base still displayed, find the
volume of the cone.
π
14
× 16 × ( )1 3÷ ENTER
The volume of the cone is approximately 3,282 cm3
.
B = πr2
V = Bh
1
3

A sphere is the set of points in three
dimensions that are a fixed distance from
a given point, the center. A plane that
intersects a sphere through its center
divides the two halves or hemispheres.
The edge of a hemisphere is a great
circle.

The volume of a hemisphere is exactly
halfway between the volume of a cone
and a cylinder with the same radius r and
height equal to r.

Additional Example 1: Finding the Volume of a
Sphere
Find the volume of a sphere with radius
12 cm, both in terms of π and to the
nearest tenth. Use 3.14 for π.
= 2304π cm3
≈ 7,234.6 cm3
Volume of a sphere
Substitute 12 for r.
4
3
V = πr3
= π(12)34
3

Find the volume of a sphere with radius 3
m, both in terms of π and to the nearest
tenth. Use 3.14 for π.
= 36π m3
≈ 113.0 m3
Volume of a sphere
Substitute 3 for r.
4
3
V = πr3
= π(3)34
3

Standard Lesson Quiz
Lesson Quizzes
Lesson Quiz for Student Response Systems

Lesson Quiz: Part I
Find the volume of each figure to the nearest
78.5 in3
6.3 m3
1. the triangular pyramid
2. the cone

Lesson Quiz: Part 2
Find the volume of each figure to the nearest
Yes; the volume is one-third the product of
the base area and the height. So if you triple
the height, the product would be tripled.
3. Explain whether tripling the height of a square
pyramid would triple the volume.

Lesson Quiz: Part 3
Find the volume of each sphere, both in
terms of π and to the nearest tenth. Use 3.14
for π.
4. r = 4 ft
5. d = 6 m 36π m3
, 113.0 m3
85.3π ft3
, 267.8 ft3
6. A basketball has a circumference of 29 in. To
the nearest cubic inch, what is its volume?
412 in3

1. Identify the volume of the triangular pyramid
rounded to the nearest tenth.
A. 16 m3
B. 16.5 m3
C. 17 m3
D. 17.5 m3

2. Identify the volume of the cone rounded to the
nearest tenth. Use 3.14 for π.
A. 183.2 in3
B. 176.5 in3
C. 167.2 in3
D. 128.2 in3

3. Explain whether doubling the length of the following
rectangular pyramid would double the volume.
A. Yes; the volume is one-third the product of the
base area and the height. So if you double the length
of the base, the product would be doubled.
B. No; the volume is one-third the product of the base
area and the height. So if you double the length of the
base, the product would be quadrupled.

4. Find the volume of a sphere with radius 5 ft,
both in terms of π and to the nearest tenth. Use
3.14 for π.
A. 166.7π ft3
, 523.3 ft3
B. 166.7π ft3
, 576.5 ft3
C. 183.6π ft3
, 576.5 ft3
D. 183.6π ft3
, 523.3 ft3

5. Find the volume of a sphere with a diameter of 8 m,
both in terms of π and to the nearest tenth. Use 3.14
for π.
A. 682.7π m3
, 2143.6 m3
B. 682.7π m3
, 267.9 m3
C. 85.3π m3
, 2143.6 m3
D. 85.3π m3
, 267.9 m3

6. A decorative lamp in the form of a sphere has a
circumference of 32 inches. To the nearest cubic inch,
what is its volume?
A. 554 in3
B. 562 in3
C. 571 in3
D. 580 in3

Hmh alg1 mod7 2&3 vol pyrds-cones-spheres-2016

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