Regular polygons have all sides equal and all angles equal. Below is an example of a 5 sided regular polygon also called a 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 = 360^{o} / 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 (180**^{o} / n)

and

**R = (x / 2) csc (180**^{o} / 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 (360^{o} / n)

**
Area of polygon = n * area of triangle AOB
**

= (1/2) n R ^{2} sin (360^{o} / 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 (180^{o} / n) ]

= (1 / 4) n x^{2} cot (180^{o} / n)
**Formula 4**

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

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

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

= n r^{2} tan (180^{o} / 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 __

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.