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Geometry
12.4 Volume of Prisms and Cylinders
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Geometry 12.4 Volume of Prisms and Cylinders 2Monday, May 5, 2:51
Goals
Find the volume of prisms. Find the volume of cylinders. Solve problems using volume.
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Geometry 12.4 Volume of Prisms and Cylinders 3Monday, May 5, 2:51
Volume
The number of cubic units contained in a solid.
Measured in cubic units. Basic Formula:
V = Bh B = area of the base, h = height
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Geometry 12.4 Volume of Prisms and Cylinders 4Monday, May 5, 2:51
Cubic Unit
11
1
V = 1 cu. unit
ss
s
V = s3
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Geometry 12.4 Volume of Prisms and Cylinders 5Monday, May 5, 2:51
Cavalieri’s Principle
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Geometry 12.4 Volume of Prisms and Cylinders 6Monday, May 5, 2:51
B
BB
hh
h
Prism: V = Bh
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Geometry 12.4 Volume of Prisms and Cylinders 7Monday, May 5, 2:51
Cylinder: V = r2h
B
h
r
h
V = Bh
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Geometry 12.4 Volume of Prisms and Cylinders 8Monday, May 5, 2:51
Example 1 Find the volume.
10
8
3
Triangular Prism
V = Bh
Base = 40
V = 40(3) = 120
Abase = ½ (10)(8) = 40
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Geometry 12.4 Volume of Prisms and Cylinders 9Monday, May 5, 2:51
Example 2 Find the volume.
12
10
V = Bh
The base is a ?
Hexagon
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Geometry 12.4 Volume of Prisms and Cylinders 10Monday, May 5, 2:51
Example 2 Solution
12
10
12
?12
?
?
6
12
12 6 3 72
216 3
374.1
A ap
6 3
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Geometry 12.4 Volume of Prisms and Cylinders 11Monday, May 5, 2:51
Example 2 Solution
12
10
374.1
V = Bh
V = (374.1)(10)
V 3741
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Geometry 12.4 Volume of Prisms and Cylinders 12Monday, May 5, 2:51
Example 3A soda can measures 4.5 inches high and the diameter is 2.5 inches. Find the approximate volume.
V = r2h
V = (1.252)(4.5)
V 22 in3
(The diameter is 2.5 in. The radius is 2.5 ÷ 2 inches.)
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Geometry 12.4 Volume of Prisms and Cylinders 13Monday, May 5, 2:51
Example 4A wedding cake has three layers.
The top cake has a diameter of 8 inches, and is 3 inches deep.
The middle cake is 12 inches in diameter, and is 4 inches deep.
The bottom cake is 14 inches in diameter and is 6 inches deep.
Find the volume of the entire cake, ignoring the icing.
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Geometry 12.4 Volume of Prisms and Cylinders 14Monday, May 5, 2:51
Example 4 Solution
8
12
14
3
4
6
r = 4
r = 6
r = 7
VTop = (42)(3) = 48 150.8 in3
VMid = (62)(4) = 144 452.4 in3
VBot = (72)(6) = 294 923.6 in3
486 1526.8 in3
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Geometry 12.4 Volume of Prisms and Cylinders 15Monday, May 5, 2:51
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Geometry 12.4 Volume of Prisms and Cylinders 16Monday, May 5, 2:51
Example 5A manufacturer of concrete sewer pipe makes a pipe segment that has an outside diameter (o.d.) of 48 inches, an inside diameter (i.d.) of 44 inches, and a length of 52 inches. Determine the volume of concrete needed to make one pipe segment.
44
48
52
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Geometry 12.4 Volume of Prisms and Cylinders 17Monday, May 5, 2:51
Example 5 SolutionStrategy:
Find the area of the ring at the top, which is the area of the base, B, and multiply by the height.
View of the Base
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Geometry 12.4 Volume of Prisms and Cylinders 18Monday, May 5, 2:51
Example 5 SolutionStrategy:
Find the area of the ring at the top, which is the area of the base, B, and multiply by the height.
Area of Outer Circle:
Aout = (242) = 576
Area of Inner Circle:
Ain = (222) = 484
Area of Base (Ring):
ABase = 576 - 484 = 92
44
48
52
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Geometry 12.4 Volume of Prisms and Cylinders 19Monday, May 5, 2:51
Example 5 Solution V = Bh
ABase = B = 92
V = (92)(52)
V = 4784
V 15,029.4 in3
44
48
52
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Geometry 12.4 Volume of Prisms and Cylinders 20Monday, May 5, 2:51
Example 5 Alternate Solution
Vouter = (242)(52)
Vouter = 94,096.98
Vinner = (222)(52)
Vinner = 79,067.60
V = Vouter – Vinner
V 15,029.4 in3
44
48
52
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Geometry 12.4 Volume of Prisms and Cylinders 21Monday, May 5, 2:51
Example 6
A metal bar has a volume of 2400 cm3. The sides of the base measure 4 cm by 5 cm. Determine the length of the bar.
4
5L
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Geometry 12.4 Volume of Prisms and Cylinders 22Monday, May 5, 2:51
Example 6 Solution
Method 1 V = Bh B = 4 5 = 20 2400 = 20h h = 120 cm
Method 2 V = L W H 2400 = L 4
5 2400 = 20L L = 120 cm
4
5L
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Geometry 12.4 Volume of Prisms and Cylinders 23Monday, May 5, 2:51
Example 7
3 in
V = 115 in3
A 3-inch tall can has a volume of 115 cubic inches. Find the diameter of the can.
2
2
2
2
2
115 3
115 9.42
115
9.42
12.21
12.21
3.5
V r h
r
r
r
r
r
r
Diameter = 7
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Geometry 12.4 Volume of Prisms and Cylinders 24Monday, May 5, 2:51
Summary
The volumes of prisms and cylinders are essentially the same:
V = Bh & V = r2h where B is the area of the base, h is the
height of the prism or cylinder. Use what you already know about area of
polygons and circles for B.
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Geometry 12.4 Volume of Prisms and Cylinders 25Monday, May 5, 2:51
V = Bh V = r2h
B
h h
r
Add these to your formula sheet.
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Geometry 12.4 Volume of Prisms and Cylinders 26Monday, May 5, 2:51
Which Holds More?
(3.2)(1.6)(4)
20.48
V
3.2 in 1.6 in
4 in 4.5 in
2.3 in
This one!
2
2.34.5
2
18.7
V
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Geometry 12.4 Volume of Prisms and Cylinders 27Monday, May 5, 2:51
What would the height of cylinder 2 have to be to have the same volume as cylinder 1?
r = 4
h
r = 3
8#1#2
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Geometry 12.4 Volume of Prisms and Cylinders 28Monday, May 5, 2:51
Solution
24 8
128
V
r = 4
8#1
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Geometry 12.4 Volume of Prisms and Cylinders 29Monday, May 5, 2:51
Solution
h
r = 3
#2
2128 3
128
914.2
h
h
h