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10-3 Volumes of Prisms and Cylinders

From the Box Volume Formula and Cavalieri’s Principle, volume formulas for any cylindrical solids can be deduced.

10-3 Volumes of Prisms and Cylinders

Volume = B h = (50 * 120) (h) = 6000 h yd3

Volume = B h = (150 * 360) (h) = 54,000 h ft3

10-3 Volumes of Prisms and Cylinders

Suppose a prism has a base that is the size of a basketball court, 31 yards by 17 yards. If the prism contains the amount of oil that is mentioned in Example 1 (20.7 × 106 barrels), what is the height of the prism?

(Remember: 42 gal per barrel and 7.43 gal per foot3.)

10-3 Volumes of Prisms and Cylinders

- Volume Postulate

e. Cavalieri’s Principle

Let I and II be two solids included between parallel planes. If every plane P parallel to the given planes intersects I and II in sections with the same area, the Volume(I) = Volume(II).

10-3 Volumes of Prisms and Cylinders

- Since the volume of II is Bh,
- And since areas X, Y and Z are all equal,
- And since the heights of the three volumes are all equal,
- According to Cavalieri’s Principle, all three prisms have equal volumes (Bh).

10-3 Volumes of Prisms and Cylinders

Prism-Cylinder Volume Formula

The volume V of any prism or cylinder is the product of its height h and the area B of its base.

V = Bh

10-3 Volumes of Prisms and Cylinders

The base of an oblique rectangular prism is 3 units by 6 units, and its height is 15 units. Find its volume.

10-3 Volumes of Prisms and Cylinders

A hexagonal prism has two bases: base I and base II. Base I is in plane M, and base II is in plane N. If base I is not changed, how can base II be moved so that the volume of the prism does not change?

10-3 Volumes of Prisms and Cylinders

Using Cavalieri’s Principle, a cylinder has a radius of 5 centimeters and a square prism has a base edge of 8.86227 centimeters. To the nearest thousandth, what is the area of the base of the cylinder?

What is the area of the base of the prism?

For every slice of the prism and the cylinder that is parallel to the bases, what is the area of that slice?

10-3 Volumes of Prisms and Cylinders

A rectangular prism A with height 10 centimeters has a volume of 250 square centimeters. A triangular prism B has its bases in the same planes as prism A, and prisms A and B have the same volume. A plane parallel to the bases cuts a slice of each prism.

What is the area of the slice for prism A?

For prism B?

10-3 Volumes of Prisms and Cylinders

The diagram below shows an oblique cylinder with a radius of 4 cm that just fits inside an oblique square prism. The area of the shaded region of the area of the base is equal to the base of an oblique triangular prism. All the solids have bases in parallel planes that are 15 cm apart. What is the volume of the triangular prism?

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