Regular Polygon Definition - Math Open Reference

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Regular Polygon(also Equiangular Polygon)

Definition: A polygon that has all sides equal and all interior angles equal

Try this Adjust the polygon below by dragging any orange dot. Note that the length of all sides remains the same and the interior angles remain the same.

(If there is no image below, see support page.)

Regular polygons tend to look like someone tried to make a circle out of some straight lines. In fact, if you have a polygon with very many sides, it looks a lot like a circle from a distance. The sides and vertices are evenly spread around a central point, and regular polygons are convex - all the vertices point 'outwards'.

In the figure above, you can change the number of sides using the controls on the right of the polygon. An interesting thing to note is that once the number of sides is quite high (say more than 17) the shape begins to look a lot like a circle. In fact, as you keep adding sides, the area of the polygon, the area of its incircle, and the area of its circumcircle all converge on the same value.

Attributes

Vertex	The vertex (plural: vertices) is a corner of the polygon. In any polygon, the number of sides and vertices are always equal.
Side	The sides are the straight line segments that make up the polygon. See Sides of a Regular Polygon for more information and formulas used to calculate their length.
Apothem (inradius)	The apothem of a regular polygon is the line from the center to the midpoint of a side. (or, the length of that line). It is also the radius of the incircle. See Apothem of a Regular Polygon
Radius (circumradius)	The radius is the distance from the center to any vertex. It is also the radius of the polygon's circumcircle, the circle that passes through every vertex. See Circumcircle of a regular polygon
Central Angle	The angle at the center of the polygon made by two adjacent radius lines. See Central Angles of a Regular Polygon
Area	See Area of a Regular Polygon
Perimeter	By definition, all sides are the same length, so the perimeter is simply the length of a side times the number of sides.

Being equilateral is not enough

For a polygon to be 'regular' it must have all sides the same length and all interior angles the same. The figure on the right is actually an example of an equilateral polygon since it has all sides the same length, but it is not a regular polygon because its interior angles are not all the same. See Irregular Polygon.

Related polygon topics

General

Types of polygon

Area of various polygon types

Perimeter of various polygon types

Angles associated with polygons

Named polygons