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Geometry Volume of Pyramids and Cones

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Page 1: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

Geometry

Volume of Pyramids and Cones

Page 2: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Goals Find the volume of pyramids and cones. Solve problems using volume.

Skip Intro

Page 3: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Pyramids

Page 4: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Pyramid

Polyhedron with a polygon base. Lateral faces are triangles. Lateral faces meet at a common vertex. Regular pyramid: base is a regular

polygon. (This is what we will discuss.)

Page 5: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Square PyramidVertex

Altitude (h) Slant Height (l)

Page 6: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Tempe City Hall

Page 7: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Governor G. Hunt (1859 – 1934)

Page 8: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Luxor Hotel, Las Vegas NV

Pyramid Arena, Memphis TN

Page 9: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Volume of a PyramidVolume of the Prism

V = BhVolume of the Pyramid

13V Bh

Page 10: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Volume of a ConeVolume of the Cylinder

V = r2hVolume of the Cone

213V r h

Page 11: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Example 1 Find the volume.

8

6

8

13

213

13

8 6

64 6

128

V Bh

Page 12: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Example 2 Find the volume.

10

10

12

Page 13: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Example 2 Solution

10

10

12?

5

h

Page 14: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Example 2 Solution

10

10

12

5

Solve for h:2 2 2

2

2

5 12

25 144

119

119

10.91

h

h

h

h

h

10.91h

Page 15: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Example 2 Solution

10

10

12

5

Solve for V:

10.91 13

13 100 10.91

363.7

V Bh

Page 16: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Example 3 Find the volume of the pyramid with a regular hexagonal base.

h = 15

6

Page 17: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Example 3Solution

h = 15

6

Area of the base:

6

3

12

12 3 3 36

54 3

A ap

3 3

Page 18: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Example 3Solution

54 3h = 15

6

Area of the base:

1 13 3 54 3 15 270 3 467.65V Bh

Page 19: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Example 4 Find the volume of the right circular cone.

213

213 5 12

100

314

V r h

5

12

Page 20: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Example 5 Find the volume of the right circular cone.

6

15

Page 21: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Example 5

Solution

6

15h

Find h:2 2 2

2

2

6 15

36 225

189

189

13.75

h

h

h

h

h

Page 22: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Example 5

Solution

6

1513.75

Find the volume:

213 6 (13.75)

165

518.36

V

Page 23: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Summary

The volume of a pyramid is 1/3 the volume of a prism with the same measurements.

The volume of a cone is 1/3 the volume of a cylinder with the same measurements.

Page 24: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

h

r

h

13V Bh 21

3V r h

Add these to your formula sheet.

Page 25: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Find the volume of each figure.

213

213 5 19.36

506.97

V r h

5

2012

14

14

19.36

13

213 14 12

784

V Bh

Page 26: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Final Problem

The base of a square pyramid measures 12 ft on each side, and the height of the pyramid is 15 ft.

A cone has a diameter of 12 ft and has the same volume as the square pyramid. What is the height of the cone?

Page 27: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Solution

13

213 12 15

144 15

3720

V Bh

21

3

213720 6

720

1

12

9

720

12

V r h

h

h

h

PyramidCone

Page 28: Geometry Volume of Pyramids and Cones. August 21, 2015 Goals Find the volume of pyramids and cones. Solve problems using volume. Skip Intro

April 19, 2023

Practice Problems