VOLUMES OF CYLINDERS, PYRAMIDS, AND CONES ADAPTED FROM WALCH EDUCATION

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 : VOLUMES OF CYLINDERS, PYRAMIDS, AND CONES 2

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 : VOLUMES OF CYLINDERS, PYRAMIDS, AND CONES 3

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 : VOLUMES OF CYLINDERS, PYRAMIDS, AND CONES 4

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

A POLYGONAL PRISM WITH 200 SIDES : VOLUMES OF CYLINDERS, PYRAMIDS, AND CONES 6

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 : VOLUMES OF CYLINDERS, PYRAMIDS, AND CONES 7

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 : VOLUMES OF CYLINDERS, PYRAMIDS, AND CONES 8

9 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: 3.5.2: VOLUMES OF CYLINDERS, PYRAMIDS, AND CONES 10

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

A PYRAMID WITH 100 SIDES 3.5.2: VOLUMES OF CYLINDERS, PYRAMIDS, AND CONES 12

THANKS FOR WATCHING! MS. DAMBREVILLE