Regular Polygons

A tutorial to explore the formulas and other properties of regular polygons, inscribed and circumscribed circles. An html 5 applet may also used for an interactive tutorial. There is also a calculator for regular polygons.

Regular polygons have all sides equal and all angles equal. Below is an example of a 5 sided regular polygon also called a pentagon.

5 sided polygon or pentagon


where x is the side of the pentagon, r is the radius of the inscribed circle and R is the radius of the circumscribed circle.

Let us develop formulas to find the area of an n sided regular polygons as a function of x, r and R. We shall follow the following route: Find the area of one triangle, such as triangle BOC, and multiply it by n ,the number of sides of the polygon, to find the total area of the polygon.

Relationship between x, r and R.

Let t be angle BOC.

t = 360o / n

From trigonometry of right triangles, we have

tan(t / 2) = (x / 2) / r      and      sin (t / 2) = (x / 2) / R

which gives r and R in term of x as follows

r = (x / 2) cot (180o / n)


and

R = (x / 2) csc (180o / n)

Formula 1

The area of triangle BOC = (1/2) x r

Area of polygon = n * area of triangle BOC

= (1/2) n x r

Formula 2

Another possible formula for the area of triangle BOC in terms of R is

Area of triangle BOC = (1/2) sin ( t ) R 2

= (1/2) R 2 sin (360o / n)

Area of polygon = n * area of triangle BOC

= (1/2) n R 2 sin (360o / n)

Formula 3

Another formula may be obtained if r found above is substituted in formula 1.

Area of polygon = (1/2) n x r

= (1/2) n x [ (x / 2) cot (180o / n) ]

= (1 / 4) n x2 cot (180o / n)

Formula 4

Another formula may be obtained if x in r = (x / 2) cot (180o / n) is substituted in formula 1.

Area of polygon = (1/2) n x r

= (1/2) n [ 2 r tan (180o / n) ] r

= n r2 tan (180o / n)

names of polygons according to the number of sides

number of sides name
3 equilateral triangle
4 square
5 pentagon
6 hexagon
7 heptagon
8 octagon
9 nonagon
10 decagon
11 undecagon
12 dodecagon

Interactive Tutorial

n =
3+25

>

1 - Press the button "draw" to start.

2 - In this applet, the radius of the circumscribed circle is constant. The number n of the sides of the regular polygon may be changed using the slider or by typing a positive integer greater than 3.

4 - When n increases, what happens to the three areas: that of the circumscribed circle, the polygon and the inscribed circle? See problem 4 in
polygons problems.



More geometry references

Geometry Tutorials, Problems and Interactive Applets.