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

Volume of Pyramids and ConesVolume of Pyramids, Cones & Spheres

Warm UpWarm UpProblem of the DayProblem of the DayLesson PresentationLesson PresentationLesson QuizzesLesson Quizzes

Volume of Pyramids and ConesVolume of Pyramids, Cones & SpheresWarm Up1. 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 .

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 .

39 m2

753.6 in2


Problem of the Day #1A 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 #2Find 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.


Vocabularypyramidconespherehemispheregreat circle


Additional Example 1A: Finding the Volume of Pyramids and Cones

Find the volume of the figure. Use 3.14 for .

13V = • 14 • 6

V = 28 cm3

V = Bh13

B = (4 • 7) = 14 cm2 12


Additional Example 1B: Finding the Volume of Pyramids and Cones

13V = • 9 • 10

V = 30 94.2 in3

V = Bh13

B = (32) = 9 in2

Use 3.14 for .


Volume of Pyramids and ConesVolume of Pyramids, Cones & SpheresCheck It Out: Example 1A

13V = • 17.5 • 7

V 40.8 in3

V = Bh13

B = (5 • 7) = 17.5 in2 12

5 in.

7 in.

7 in.



13V = • 9 • 7

V = 21 65.9 m3

V = Bh13

B = (32) = 9 m2

Use 3.14 for .

Check It Out: Example 1B

7 m

3 m


Volume of Pyramids and ConesVolume of Pyramids, Cones & SpheresAdditional 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.

Volume of Pyramids and ConesVolume of Pyramids, Cones & SpheresCheck 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

13

V = r2h13

13

13= (22)5

20.93 m3

13

V = r2 (2h)

= (22)(2•5) = (2 • 2)2(5)

V = (2r)2h

41.87 m3 83.73 m3

13

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.

Volume of Pyramids and ConesVolume of Pyramids, Cones & SpheresAdditional 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 13V = (3025)(24)

V = 24,200 m3

A = bh

V = Bh13

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 13V = (2304)(12)

V = 9216 m3

A = bh

V = Bh13

Find the volume of a pyramid with a height of 12 m and a base with 48 m sides.

Volume of Pyramids and ConesVolume of Pyramids, Cones & SpheresAdditional 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.

15

64 ( )1 3÷ ENTER

The volume of the cone is approximately 15,080 cm3.

B = r2

V = Bh13


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.

14

16 ( )1 3÷ ENTER

The volume of the cone is approximately 3,282 cm3.

B = r2

V = Bh13


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 SphereFind the volume of a sphere with radius 12 cm, both in terms of and to the nearest tenth. Use 3.14 for .

= 2304 cm3 7,234.6 cm3

Volume of a sphere

Substitute 12 for r.

43 V = r3

= (12)343


Find the volume of a sphere with radius 3 m, both in terms of and to the nearest tenth. Use 3.14 for .

= 36 m3 113.0 m3

Volume of a sphere

Substitute 3 for r.

43 V = r3

= (3)343


Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

Volume of Pyramids and ConesVolume of Pyramids, Cones & SpheresLesson Quiz: Part I

Find the volume of each figure to the nearest tenth. Use 3.14 for .

78.5 in3

6.3 m3

1. the triangular pyramid

2. the cone

Volume of Pyramids and ConesVolume of Pyramids, Cones & SpheresLesson Quiz: Part 2

Find the volume of each figure to the nearest tenth. Use 3.14 for .

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.

Volume of Pyramids and ConesVolume of Pyramids, Cones & SpheresLesson Quiz: Part 3

Find the volume of each sphere, both in terms of and to the nearest tenth. Use 3.14 for .4. r = 4 ft

5. d = 6 m 36m3, 113.0 m3

85.3ft3, 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 .

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 and to the nearest tenth. Use 3.14 for .

A. 166.7ft3, 523.3 ft3

B. 166.7ft3, 576.5 ft3

C. 183.6ft3, 576.5 ft3

D. 183.6ft3, 523.3 ft3



5. Find the volume of a sphere with a diameter of 8 m, both in terms of and to the nearest tenth. Use 3.14 for .

A. 682.7m3, 2143.6 m3

B. 682.7m3, 267.9 m3

C. 85.3m3, 2143.6 m3

D. 85.3m3, 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. 554in3

B. 562in3

C. 571in3

D. 580in3


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

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