# Definition:Polygon/Regular

## Definition

A **regular polygon** is a polygon which is both equilateral and equiangular.

That is, in which all the sides are the same length, and all the vertices have the same angle:

### Center

The **center** of a **regular polygon** $P$ is defined as the point which is the center of the circumcircle of $P$.

In the above, $O$ is the **center** of the **regular polygon**.

### Long Radius

The **long radius** of a **regular polygon** $P$ is defined as the distance from the center of $P$ to one of its vertices.

In the above, the length of $OA$ is the **long radius** of the **regular polygon**.

### Apothem

The **apothem** of a **regular polygon** $P$ is defined as the perpendicular distance from the center of $P$ to one of its sides.

In the above, the length of $OM$ is the **apothem** of the **regular polygon**.

## Also known as

In Euclid's *The Elements*, a **regular polygon** is referred to as an **equilateral and equiangular polygon**.

Some sources use the word **perfect** or **symmetrical** instead of **regular**.

## Examples

Specific instances of regular polygons with specific numbers of sides are as follows:

- $3$ sides: Equilateral Triangle
- $4$ sides: Square
- $5$ sides: Regular Pentagon
- $6$ sides: Regular Hexagon
- $7$ sides: Regular Heptagon
- $8$ sides: Regular Octagon
- $9$ sides: Regular Nonagon or Regular Enneagon
- $10$ sides: Regular Decagon
- $11$ sides: Regular Hendecagon or Regular Undecagon
- $12$ sides: Regular Dodecagon

- $17$ sides: Regular Heptadecagon

The term **regular $n$-gon** is usually used nowadays to specify a regular polygon with a specific number, that is $n$, sides.

The specific name is usually invoked only in order to draw attention to the fact that such a regular polygon has a particularly interesting set of properties.

## Also see

- Results about
**regular polygons**can be found**here**.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{IV}$: The Prince of Amateurs - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.4$: Euclid (flourished ca. $300$ B.C.) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**polygon** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**polygon** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**polygon** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**regular polygon**