nLab regular polygon

 Definition

In Euclidean geometry, a regular polygon is a (simple, non self-intersecting) polygon in a Euclidean space $\mathbb{R}^n$ such that every segment? of the boundary of the polygon has the same length and every angle between two segments of the boundary of the polygon has the same angle measure. (The boundary of a regular polygon is sometimes called called a regular polygonal line)

Perimeter and area

We are using the circle constant $\tau = 2 \pi$.

Every regular polygon is the union of $n$ congruent triangles each with segments of length $r$ and $r$ respectively and an angle of $\frac{\tau}{n}$ between the two segments of length $r$. As a result, the triangles are isosceles and the altitude from the center of the regular polygon to the third segment of the triangles bisects the angle: such that the angles between the circumradius and the altitude is $\frac{\tau}{2n}$. The length of the third segment is thus given by

$$b \coloneqq r \sin\left(\frac{\tau}{2n}\right) + r \sin\left(\frac{\tau}{2n}\right) = 2 r \sin\left(\frac{\tau}{2n}\right)$$

The perimeter of a regular polygon $\mathcal{P}_n$ (strictly speaking, a regular polygonal line) with $n$ sides and circumradius $r$ is given by the sequence of functions $P_\mathcal{P}:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$

$$P_\mathcal{P}(n)(r) = n b = r (2 n) \sin\left(\frac{\tau}{2n}\right)$$

The area of a regular polygon $\mathcal{P}_n$ with $n$ sides and circumradius $r$ is given by the sequence of functions $P:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$

$$A_\mathcal{P}(n)(r) = n \left(\frac{1}{2} r b\right) = n \frac{1}{2} r (2 r) \sin\left(\frac{\tau}{2n}\right) = \frac{1}{2} r^2 (2 n) \sin\left(\frac{\tau}{2n}\right)$$

See also

• polygon

• center of a regular polygon

• circumference of a circle

• area of a circle

References

• Wikipedia, Regular polygon