Regular Polygons

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

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 OAB, 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 AOB.

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)


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

Formula 1

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

Area of polygon = n * area of triangle AOB

= (1/2) n x r

Formula 2

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

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

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

Area of polygon = n * area of triangle AOB

= (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

Your browser is completely ignoring the <APPLET> tag!

1 - Press the button above to start the applet.

2 - In this applet, the radius of the circumscribed circle is constant and equal to 3. The number n of the sides of the polygon may be changed using the slider.

3 - Use the formulas found above to check the area of the polygon for different values of n.

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.