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Regular, Irregular, Equilateral, and Equiangular Polygons

In geometry, polygons are flat, two-dimensional shapes bounded by straight lines. There are a number of ways to classify polygons, such as by number of sides or by their regularity. When classifying them by number of sides, the prefix indicates the number: triangle (3 sides), quadrilateral (4 sides), pentagon (5 sides), hexagon (6 sides), etc.

Categorizing polygons by their regularity means checking if their side lengths and interior angles are all equal. A figure whose sides all the same length is called equilateral. A figure whose angles are equal is called equianglular. And a polygon that is both equilateral and equiangular is called regular. Irregular polygons have sides and/or angles that are unequal.

It is important to note that equilateral and equiangular are independent from one another except in the case of a triangle. A polygon with more than 3 sides can have equal sides but unequal angles, and vice versa.


Equilateral Shapes

With triangles, if the sides are all equal then the angles must also be equal. The angle measure in an equilateral triangle is 60 degrees.

The special name for quadrilaterals whose sides are equal is "rhombus." A square is a particular type of rhombus.

For polygons with 5 or more sides, there are no specials names for equilateral figures. If all the sides are equal, you simply call the shape an equilateral pentagon or equilateral hexagon.

Two equilateral pentagons (black) and two equliateral hexagons (red).  Even though each shape has equal sides, they are irregular because the angles are unequal.
Two equilateral pentagons (black) and two equliateral hexagons (red). Even though each shape has equal sides, they are irregular because the angles are unequal.

Equiangular Shapes

A quadrilateral that has equal angles all around is called a rectangle. A square is a particular type of rectangle, and the only rectangle that is also a rhombus.

For 5-sided figures, in order to be equiangular all the angles must be 108 degrees. For 6-sided figures, all angles must be 120 degrees.

If an equiangular polygon has N sides, then the angles are all equal to 180 - 360/N degrees. As N grows, the angle measure of an equiangular N-gon approaches 180 degrees. For instance, an equiangular icosagon (20-sided shape) has all angles equal to 162 degrees, since 180 - 360/20 = 162. An equiangular hexacontagon (60 sides) has all angle measures equal to 174 degrees.

Two equiangular pentagons (black) and two equiangular hexagons (red). These are irregular polygons since their sides have unequal lengths.
Two equiangular pentagons (black) and two equiangular hexagons (red). These are irregular polygons since their sides have unequal lengths.

Regular Polygons

Regular polygons are the preferred shape for signs, decorative objects, and architecture since they are symmetric and pleasing to the eye. Except for "square," there are no special names for regular polygons. Sometimes a regular octagon is called a stop-sign shape.

There are many ways to construct a regular polygon with N sides using a ruler, compass, and protractor. An easy method is to start with a circle and divide the disc into N wedges of equal angle measure. Then connect the adjacent points along the circumference of the circle to create the polygon. For example, to draw a regular pentagon you would start by dividing the circle in to 5 wedges whose angles equal 360/5 = 72 degrees.

Regular polygons
Regular polygons

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Comments 5 comments

SidKemp profile image

SidKemp 3 years ago from Boca Raton, Florida (near Miami and Palm Beach)

Thanks for this introduction (or reminder) to the categorization of shapes. There's another category term I'm trying to remember, can you help? Perhap it is concave and convex. In one case, any line segment drawn connecting two points within the polygon remains entirely within the polygon. That is, the polygon has no indentations. In the other case, the polygon has indentations, and one can draw a straight line from one point in the polygon to another, and part of that line will be outside the polygon.

Voted up and interesting.


calculus-geometry profile image

calculus-geometry 3 years ago from Germany Author

I believe that is called a convex polygon. My favorite are cyclic polygons, you can construct a circle so that all the vertices lie on its circumference.


SidKemp profile image

SidKemp 3 years ago from Boca Raton, Florida (near Miami and Palm Beach)

Yes, that's it - convex vs. concave polygons. One trick of your recent hexagon puzzle was that it had concave hexagons. I think all regular polygons are cyclic polygons. (Picture progressively inscribing or circumscribing a circle with an equilateral triangle, a square, a pentagon, etc. Are there any cyclic polygons that are not regular polygons, or do "cyclic" and "regular" end up being synonyms?


calculus-geometry profile image

calculus-geometry 3 years ago from Germany Author

As you said, all regular polygons are cyclic, but not all cyclic polygons are regular. To construct an irregular cyclic polygon, just take a circle and mark N points on the circumference so that they are not evenly spaced. Then connect the adjacent points and you'll get an irregular N-gon.

All rectangles are cyclic and the diagonal of the rectangle corresponds to the diameter of the circumscribing circle. All triangles are also cyclic, since three points uniquely define a circle.

Thanks for the great questions, Sid!


SidKemp profile image

SidKemp 3 years ago from Boca Raton, Florida (near Miami and Palm Beach)

Got it, I was unconsciously picturing only equilateral polygons when picturing cyclic polygons.

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