Geometry: Pyramids

Pyramids

A geometry book would not be complete if it did not include more than a passing reference to pyramids. The Egyptians have made pyramids famous, and people have traveled many miles to see these impressive structures. Engineers and archeologists are equally impressed with the design and construction of the pyramids.

You probably have an idea of what a typical pyramid looks like. It has a square base and four triangular sides rising up to join at a single point. It might come as a surprise to learn that from a mathematician's perspective, there are many kinds of pyramids. To construct a general pyramid, start with a polygon (it doesn't have to be a square, or even four-sided!) in a plane and a point above the polygon. Connect each vertex of the polygon to this point and voil! You have a pyramid. The polygon that you started with is called the base of the pyramid, and the point is called the apex of the pyramid. I have drawn a traditional pyramid and a pentagonal pyramid for you in Figure 21.3. Notice that the lateral faces of a pyramid are triangles.

Figure 21.3A square pyramid and a pentagonal pyramid.

Just as it is with prisms, the name of a pyramid is determined by the shape of its base. There are hexagonal pyramids and triangular pyramids. Try to reason out the polygon that forms the base of each of these pyramids. There are also regular pyramids: pyramids whose base is a regular polygon and whose lateral edges are all congruent.