Equilateral & Equiangular Polygons: Definition & Examples

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

In this lesson, we will discuss definitions and examples of special polygons--geometric figures with straight sides. Equilateral ones have matching side lengths. Equiangular ones have matching angles. Some polygons have both.

What are Equilateral and Equiangular Polygons?

Imagine an ice cube--a perfectly square frozen block of ice, sitting on the desk. If you look at it, straight on, from the top, you'll see that it looks like a square. Its sides all measure the same length. Its edges all come together to make perfectly square corners, 90 degrees, just like the corners of a typical house. The top of that ice cube--if it's a perfectly square one--forms one kind of equilateral and equiangular polygon.

A polygon (from Greek polugonosis, meaning 'many angles') is a geometric figure with straight sides. We see them all over--squares, rectangles, pentagons, octagons, and all sorts of other types. Houses, office buildings, streets, signs--all of these typically have polygons, straight sides, in their shapes. An important thing to remember, though, is that polygons do NOT have curved surfaces. Most cars, for example, have curves in their windshields, circular tires, curving fenders, etc. Those shapes--shapes that include curves--are not polygons.

So what's an equilateral polygon? ('Equilateral' comes from Latin aequilateralus, meaning 'equal sides'.) Well, like the ice cube we were visualizing, an equilateral polygon not only has straight sides, it also has sides that are all the same length. For example, a five-sided polygon like the shape of the Pentagon--the United States Department of Defense Headquarters in Virginia--has five sides that are all the same length. Of course, not every pentagon you come across will have sides that are over 900 feet long!

The Pentagon is shaped like a five-sided polygon that is both equilateral and equiangular
Pentagon building

An equiangular polygon (from Latin aequi meaning 'equal' and angularis meaning 'having corners or sides') is a figure with straight sides, like other polygons, but in this figure every internal angle has the same measure. For example, rooms in a typical house usually have square corners--that is, the measure of every corner is 90 degrees. You may have long rooms and short rooms, wide rooms and narrow rooms, but they usually have square corners. Any room that has straight walls and all square corners is an equiangular polygon.

Examples of Equilateral and Equiangular Polygons

There are as many possible types of equilateral and equiangular polygons as there are possible numbers of sides on a polygon. Infinite combinations. Some of the more common ones are shown below:

Equilateral and Equiangular Three-Sided Polygons (Triangles)

Because of how rigid and structured a triangle is, all equilateral triangles are also equiangular. Every equilateral triangle has three 60-degree angles inside the triangle, and three 120-degree angles outside the triangle.

All equilateral triangles are also equiangular

Equilateral and Equiangular Four-Sided Polygons (Quadrilaterals)

Equilateral quadrilaterals (four-sided polygons) all have four equal sides, regardless of the angles between the sides. Also called a rhombus, these figures are either a square, in which case all four angles also are equal, or a non-square rhombus.

A rhombus that is not also a square has equal angles in opposite corners

A square is equilateral and equiangular

Other Equiangular and Equilateral Polygons

The equilateral pentagon will also be equiangular, unless one corner is pushed in, making it concave

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