KEY CONCEPTS

• The formula for finding the volume of a prism is

V = length • width • height.

• can also be shown as V = area of base • height.

• Bonaventura Cavalieri, an Italian mathematician,

formulated Cavalieri’s Principle. This principle states that

the volumes of two objects are equal if the areas of their

corresponding cross sections are in all cases equal.

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CAVALIERI’S PRINCIPLE

• This principle is illustrated by the diagram below. A

rectangular prism has been sliced into six pieces and

is shown in three different ways.

• The six pieces maintain their same volume regardless

of how they are moved.

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A PRISM, A PRISM AT AN OBLIQUE ANGLE, AND A

CYLINDER

• The three objects meet the two criteria of Cavalieri’s Principle. First, the objects have the same height. Secondly, the areas of the objects are the same when a plane slices them at corresponding heights. Therefore, the three objects have the same volume.

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CYLINDERS

• A cylinder has two bases that are parallel. This is also

true of a prism.

• The formula for finding the volume of a cylinder is

• A cylinder can be thought of as a prism with an infinite

number of sides

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A POLYGONAL PRISM WITH 200 SIDES.

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KEY CONCEPTS, CONTINUED

• A square prism that has side lengths of will

have a base area of on every

plane that cuts through it. The same is true of a

cylinder, which has a radius, r.

• The base area of the cylinder will be . This

shows how a square prism and a cylinder can

have the same areas at each plane.

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r p

r p · r p = pr 2

pr 2

PYRAMIDS

A pyramid is a solid or hollow polyhedron object that

has three or more triangular faces that converge at a

single vertex at the top; the base may be any polygon.

• A polyhedron is a three-dimensional object that has

faces made of polygons.

• A triangular prism can be cut into three equal

triangular pyramids.

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Triangular prisms

Corresponding triangular pyramids

KEY CONCEPTS, CONTINUED

• A cube can be cut into three equal square pyramids.

• This dissection proves that the volume of a pyramid is one-third the volume of

a prism:

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V

pyramid=

1

3B · h.

CONES

A cone is a solid or hollow object that tapers from a circular

base to a point.

• A cone and a pyramid use the same formula for finding

volume.

• This can be seen by increasing the number of sides of a

pyramid.

• The limit approaches that of being a cone.

• The formula for the volume of a cone is

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A PYRAMID WITH 100 SIDES

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