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