nLab regular polygon

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 Definition

In Euclidean geometry, a regular polygon is a (simple, non self-intersecting) polygon in a Euclidean space n\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 τ=2π\tau = 2 \pi.

Every regular polygon is the union of nn congruent triangles each with segments of length rr and rr respectively and an angle of τn\frac{\tau}{n} between the two segments of length rr. 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 τ2n\frac{\tau}{2n}. The length of the third segment is thus given by

brsin(τ2n)+rsin(τ2n)=2rsin(τ2n)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 𝒫 n\mathcal{P}_n (strictly speaking, a regular polygonal line) with nn sides and circumradius rr is given by the sequence of functions P 𝒫:()P_\mathcal{P}:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})

P 𝒫(n)(r)=nb=r(2n)sin(τ2n)P_\mathcal{P}(n)(r) = n b = r (2 n) \sin\left(\frac{\tau}{2n}\right)

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

A 𝒫(n)(r)=n(12rb)=n12r(2r)sin(τ2n)=12r 2(2n)sin(τ2n)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

References

Last revised on May 17, 2022 at 08:12:15. See the history of this page for a list of all contributions to it.