geometry volume of pyramids and cones. august 21, 2015 goals find the volume of pyramids and cones....

Geometry

Volume of Pyramids and Cones

April 19, 2023

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

Skip Intro

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Pyramids

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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.)

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Square PyramidVertex

Altitude (h) Slant Height (l)

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Tempe City Hall

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Governor G. Hunt (1859 – 1934)

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Luxor Hotel, Las Vegas NV

Pyramid Arena, Memphis TN

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Volume of a PyramidVolume of the Prism

V = BhVolume of the Pyramid

13V Bh

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Volume of a ConeVolume of the Cylinder

V = r2hVolume of the Cone

213V r h

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Example 1 Find the volume.

8

6

8

13

213

13

8 6

64 6

128

V Bh

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Example 2 Find the volume.

10

10

12

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Example 2 Solution

10

10

12?

5

h

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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

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Example 2 Solution

10

10

12

5

Solve for V:

10.91 13

13 100 10.91

363.7

V Bh

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

h = 15

6

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Example 3Solution

h = 15

6

Area of the base:

6

3

12

12 3 3 36

54 3

A ap

3 3

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Example 3Solution

54 3h = 15

6

Area of the base:

1 13 3 54 3 15 270 3 467.65V Bh

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Example 4 Find the volume of the right circular cone.

213

213 5 12

100

314

V r h

5

12

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Example 5 Find the volume of the right circular cone.

6

15

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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

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Example 5

Solution

6

1513.75

Find the volume:

213 6 (13.75)

165

518.36

V

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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.

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h

r

h

13V Bh 21

3V r h

Add these to your formula sheet.

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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

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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?

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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

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Practice Problems