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Geometric Surfaces

Pyramids and Cones

Problems

Problems

Pyramids

Another interesting kind of polyhedron is a pyramid. A pyramid is the union of a polygon with all of the segments that have one endpoint on the polygon and the other endpoint at a specified point in space that is not in the same plane as the polygon. The polygon is the base of the pyramid, and the fixed noncoplanar point is the vertex of the pyramid. The faces of the pyramid that share the vertex are the lateral faces of the pyramid, and their intersections with each other are called the lateral edges.

Figure %: A pyramid
Every lateral face of a pyramid is a triangle. The segment perpendicular to the plane of the polygon with one endpoint at the vertex and one in the plane is called the altitude of the pyramid.

A regular pyramid is a special kind of pyramid (ironic, huh?) whose base is a regular polygon and whose altitude intersects the plane of the base at the center of the base. This means that all of the lateral edges and lateral faces are congruent.

Figure %: A regular pyramid

Cones

A more general form of a pyramid is a cone. A cone is composed of a simple closed curve in a plane and all of the line segments that join that curve with a fixed point not in the plane of the curve. The simple closed curve is called the base of the cone, and the fixed noncoplanar point is the vertex. The segments that connect the base with the vertex form the lateral surface of the cone. The segment perpendicular to the plane of the base with an endpoint at the vertex and an endpoint on the plane is the altitude of the cone.

Figure %: A cone

A circular cone is a special kind of cone whose base is a circle. A right circular cone is a circular cone whose altitude intersects the plane of the circle at the circle's center.

Figure %: A right circular cone
It is easy to see the close relationship between pyramids and cones. The only difference is the base--a pyramid is a cone with a polygonal base.

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