## Regular Polygons

### Perimeter of Regular Polygons

Where $$n$$ is the number of sides, and $$s$$ is the length of a side, and $$r$$ is the outer radius (the radius of a circumscribed circle), the perimeter of a regular polygon can be calculated as follows:

$$sin\frac{π}{n}=\frac{s}{2r}$$

Where $$s=2r\,sin\frac{π}{n}$$

Perimeter $$ρ=ns$$

So $$ρ=n(2r\,sin\frac{π}{n})$$ $$=2(n\,sin\frac{π}{n})r$$

This is generally written as $$ρ=2νr$$, where $$ν$$ is the product $$ν=n\,sin\frac{π}{n}$$.

One might not think of a circle as a regular polygon, but in reality, a circle is a regular polygon with an infinite number of sides. If we were to evaluate $$ν$$ as the limit $$ν=\lim\limits_{n\to\infty}n\,sin\frac{π}{n}$$, we would get $$ν=π$$ as a result. The perimeter of a circle is then $$2νr=2πr$$.

### Area of Regular Polygons

Since a regular polygon is composed of $$n$$ isosceles triangles, each with an area of $$k=\frac{1}{2}sa$$, where $$s$$ is the length of one side, and $$a$$ is the length of the inner radius of the polygon (the radius of an incribed circle), it follows that the polygon will have an area equal to $$n(\frac{1}{2}sa)$$, where $$n$$ is the number of sides; therefore, for any regular polygon, the area $$k$$ can be calculated as:

$$k=\frac{1}{2}nsa$$ $$=\frac{1}{2}ρa$$

Also, in terms of radius $$r$$:

$$k=\frac{1}{2}(2νr)(r\,cos\frac{π}{n})$$ $$=ν\,cos\frac{π}{n}r^2$$ $$=\frac{n}{2}sin\frac{2π}{n}r^2$$

And in terms of perimeter, where $$ρ=\frac{r}{2ν}$$:

$$k=ν\,cos\frac{π}{n}(\frac{ρ}{2ν})^2$$ $$=\frac{cos\frac{π}{n}{ρ^2}}{4ν}$$ $$=\frac{ρ^2\,cos\frac{π}{n}}{4π\,sin\frac{π}{n}}$$ $$=\frac{ns^2\,cos\frac{π}{n}}{4π\,sin\frac{π}{n}}$$ $$=\frac{ns^2}{4}cot\frac{π}{4}$$ $$=\frac{1}{4}ns^2\,cot\frac{π}{n}$$

If we know the area and number of sides, we can determine that the length of a side is:

$$s=\sqrt{\frac{4k}{n\,\cot\frac{π}{n}}}$$

And if we know the side length and number of sides, we can determine the inner radius $$a$$ and outer radius $$r$$:

$$a=\frac{s^2}{2}\,cot\frac{π}{n}$$

$$r=\frac{s}{2}\,csc\frac{π}{n}$$

The area formula $$A=\frac{1}{2}ρa$$, as described before can be used for any regular polygon. A regular polygon always has sides of equal length. Below are some examples:

As long as all three sides of a triangle have equal length, then it is a regular polygon and has an area equal to:

$$A=\frac{pr}{2}$$

A square is the regular polygon with 4 sides:

$$A=\frac{pr}{2}$$
$$=4r^2$$

A hexagon has six equal sides:

$$A=\frac{pr}{2}$$

As stated before, a circle is a regular polygon with an infinite number of sides:

$$A=\frac{cr}{2}$$
$$=πr^2$$

A pentagram is also a regular polygon. Remember to use the inner radius:

$$A=\frac{pr}{2}$$

The same is true of a hexagram, or any similar shape with equal-length sides:

$$A=\frac{pr}{2}$$

A rhombus might not look like a regular polygon, but it is:

$$A=\frac{pr}{2}$$