Author:
Christopher Danielson

To generalize the procedures of constructing equilateral triangles and quadrilaterals to figures with more sides.

This packet follows up on other packets on geometric constructions (referenced and linked in the introduction) and moves to generalizing the use of a compass to determine side lengths of polygons.

Tutorial

So you know how to construct a rhombus, and you know why a compass and straightedge are able to do it. But what about other equilateral figures?

The process is the same, but there are more steps as we get more sides. An equilateral pentagon requires five congruent sides, of course, and an equilateral hexagon six, etc.

One of the challenging parts of doing constructions is that we are having to imagine sides of figures by their endpoints. We tend to think of a pentagon as having five **sides**. But when we are constructing, the sides are secondary-they come after we locate the **vertices** of the pentagon. And to make matters more abstract, the vertices lie on circles. (Why? See my packet on circles.)

As you perform the constructions in this packet, notice when you can choose *any *point on a circle and notice when you need a *special* point on a circle. It turns out that the last vertex of each equilateral polygon has to be carefully placed because it determines two side lengths at the same time. Watch for this in each construction.

NOTE: If you are just getting started with constructions, I recommend the other packets listed below as prerequisites.

Constructing an equilateral triangle

This video demonstrates construction of an equilateral pentagon that is not regular.

This video demonstrates constructions for two equilateral hexagons, both special. The first is special because it is regular (i.e. also equiangular). The second is special because it has all angles measuring 60° or multiples of 60°.

This video demonstrates the construction of an equilateral hexagon with no particular special properties.

As a general principle, a compass determines the set of points that are a common distance from the center. In constructing a regular *n*-gon, we start with two vertices at the endpoints of one side. From one of those vertices, we construct a circle using the side length as radius, we choose a point on that circle and repeat the process.

This continues until we need to locate the *n*th vertex. That last vertex needs to be the same distance from the previous one that it is from the first vertex. The last vertex needs to be at the intersection of two circles.

Along the way, we do need to be strategic about where we put our vertices. If we aren't careful, the second-to-last vertex may be too far from the first to connect with only one more vertex.