VOLUMES OF CYLINDERS, PYRAMIDS, AND CONES ADAPTED FROM WALCH
EDUCATION
Slide 3
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
Cavalieris 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. 3.5.2: VOLUMES OF CYLINDERS,
PYRAMIDS, AND CONES 2
Slide 4
CAVALIERIS 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. 3.5.2: VOLUMES OF
CYLINDERS, PYRAMIDS, AND CONES 3
Slide 5
A PRISM, A PRISM AT AN OBLIQUE ANGLE, AND A CYLINDER The three
objects meet the two criteria of Cavalieris 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. 3.5.2: VOLUMES
OF CYLINDERS, PYRAMIDS, AND CONES 4
Slide 6
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 3.5.2: VOLUMES OF CYLINDERS, PYRAMIDS, AND
CONES 5
Slide 7
A POLYGONAL PRISM WITH 200 SIDES. 3.5.2: VOLUMES OF CYLINDERS,
PYRAMIDS, AND CONES 6
Slide 8
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. 3.5.2: VOLUMES OF CYLINDERS,
PYRAMIDS, AND CONES 7
Slide 9
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.
3.5.2: VOLUMES OF CYLINDERS, PYRAMIDS, AND CONES 8
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: 3.5.2: VOLUMES OF
CYLINDERS, PYRAMIDS, AND CONES 10
Slide 12
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 3.5.2: VOLUMES OF CYLINDERS,
PYRAMIDS, AND CONES 11
Slide 13
A PYRAMID WITH 100 SIDES 3.5.2: VOLUMES OF CYLINDERS, PYRAMIDS,
AND CONES 12