Holiday Readings at eBooks.com



Regular Polygons

Perimeter of Regular Polygons

Regular Polygon
Regular Polygon

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:
Equilateral Triangle
Equilateral Triangle

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

\(A=\frac{pr}{2}\)


Square
Square

A square is the regular polygon with 4 sides:

\(A=\frac{pr}{2}\)
 \(=4r^2\)


Hexagon
Hexagon

A hexagon has six equal sides:

\(A=\frac{pr}{2}\)


Circle
Circle

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

\(A=\frac{cr}{2}\)
 \(=πr^2\)


Pentagram
Pentagram

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

\(A=\frac{pr}{2}\)


Hexagram
Hexagram

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

\(A=\frac{pr}{2}\)


Rhombus
Rhombus

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

\(A=\frac{pr}{2}\)




Geometry: Triangles