Geometry: 3-D Measurements: Volumes of Polyhedra and Spheres

In this section, we'll take a look at some formulas for calculating the volumes of some of the most common polyhedra.

Volume of a Prism

The volume of a prism is equal to the product of the area of its base and the length of its altitude; V = Bh, where B is the area of the base and h is the length of the altitude (the height). The altitude of a prism is a segment with one endpoint in one of the bases, the other endpoint in the plane that contains the other base, perpendicular to that base. It is often called the height of the prism. The area of the base is a simple calculation of the area of whichever polygon forms the base of the prism.

Volume of a Cylinder

Recall that a prism is only one special case of a cylinder. Unlike a prism, a cylinder's base can be any simple closed curve, not necessarily a polygon. The formula for the volume of a cylinder is roughly the same as that for a prism, though. The volume of a cylinder is the area of its base times the length of its altitude; V = Bh, where B is the area of the base and h is the length of the altitude (the height). Again, the altitude is the segment with one endpoint in one of the bases, the other endpoint in the plane that contains the other base, and perp endicular to that base. A circular cylinder adheres to this volume formula, but can also be written as Π times the radius squared times the height: V = Πr²h. This is only a different way to write the product of the altitude and the area of the base (since the area of a circle is derived differently than the area of a polygon..

Volume of a Pyramid

A pyramid has a slightly more complicated formula for its volume. The volume of a pyramid is equal to 1/3 the product of the area of its base and the length of its altitude. This formula is often written V = (1/3)Bh, where B is the area of the base and h is the length of the altitude (the height). This formula is especially important to know because by selecting a point inside any polyhedron as the vertex of a pyramid, that polyhedron can b e broken down into components that are all pyramids. Just as a polygon will have as many triangles as it has sides, so will a polyhedron have as many pyramids as it does faces. With this method, we can find the volume of any polyhedron by breaking it up into a number of pyramids, calculating their individual volumes, and adding those volumes together.

Volume of a Cone

The pyramid, like the prism, in only a specific case of a more general solid. All pyramids are cones with polygons for bases. A cone can have any simple closed curve as its base. The formula to find the volume of a cone is the same as that for a pyramid, however: 1/3 the product of the base's area and the altitude, or V = (1/3)Bh. When the base of a cone is a circle, the cone is a circular cone. The volume of a circular cone is (1/3)Π times the square of the radius times the length of the altitude; V = (1/3)Πr²h. Note that this is only another way to express the formula for a cone--it is a little more specific because we know a little more about this particular cone, it's base is a circle.

Volume of a Sphere

The volume of a sphere, just like its surface area, is dependent solely on its radius. The volume of a sphere is equal to (4/3)Π times the radius cubed; V = (4/3)Πr³.

Remember that the volume of a sphere and all of the other solids in this section are volumes of solids, not surfaces.