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Constructing other equilateral figures

Constructing other equilateral figures

Description:

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.

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Tutorial

Introduction

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.

Circles

Constructing an equilateral triangle

Constructing an equilateral quadrilateral I

Constructing an equilateral quadrilateral II

Constructing an equilateral pentagon

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

Constructing special equilateral hexagons

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°.

Constructing a general equilateral hexagon

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

Summary

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 nth 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.