Geometry

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Spheres

A sphere is a surface where all points are equidistant from one point, called the center. If a plane cuts a sphere, the points where they intersect form a circle. The largest circle (called a great circle) is produced when the plane passes through the center of the sphere.

The equator on Earth is a great circle. The surface area of a sphere is given by A = 4pr², its volume by V = 4/3pr³.

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Polyhedrons

A polyhedron is a figure bounded by plane surfaces. If the faces of the polyhedron are all congruent regular polygons, the polyhedron is said to be regular. It has been proven that the five regular polyhedrons—the tetrahedron (four sides), cube (six sides), octahedron (eight sides), dodecahedron (12 sides), and icosahedron (20 sides)—are the only ones possible. These five polyhedrons were known to the ancient Greek geometers. All polyhedrons (regular or not) have the remarkable property that the number of faces (the flat sides) plus the number of vertices (the angles where edges intersect) equals the number of edges plus 2. Up to relatively recent times, polyhedrons were believed to have mystic associations with natural phenomena.

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Prisms

A prism is a polyhedron that has parallel and congruent polygons, called bases, for two faces and parallelograms for all other faces. A parallelepiped is a variety of prism whose bases are parallelograms. A right prism has rectangles for sides (but not necessarily for bases). The volume of any prism is equal to the area of one of its bases times its height: V = bh.

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Pyramids

A pyramid is a polyhedron that has a polygon as its base and sides that consist of triangles having a common vertex, called the apex. A pyramid is a regular right pyramid if its base is a regular polygon and if a line joining the center of its base to its apex is perpendicular to its base. The volume of any pyramid is equal to one-third the area of its base times its height: V = €bh.

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Cylinders and Cones

A cylinder is a prism with circular bases. The formula for the volume of a cylinder is therefore the same as for a prism: A = bh. If the line connecting the centers of the two bases is perpendicular to those bases, the cylinder is a right cylinder; otherwise, it is oblique.

A cone is a pyramid with a circular base. A cone is a right cone if a line joining the center of its base to its apex is perpendicular to its base. The formula for the volume of a cone is the same as for a pyramid: V = €bh.