This page describes how to derive the formula for the
area of a regular polygon
by breaking it down into a set of *n* isosceles triangles, where *n* is the number of sides.

As shown below, a regular polygon can be broken down into a set of congruent isosceles triangles. In this case the hexagon has six of them.
If we can calculate the area of one of the triangles we can multiply by *n* to find the total area of the polygon.

If we look at one of the triangles and draw a line from the apex to the midpoint of the base it will form a right angle.
Let the length of this line be *h*. The base of the triangle is *s*, the side length of the polygon.
The whole angle at the apex is dependent on the number of sides *n*:
So the angle *t* is half that, which simplifies to
We know that the tan of an angle is opposite side over adjacent side, so
Rearranging to solve for *h*:
The area of any triangle is half the base times height, so
Which simplifies to
Finally since have *n* triangles, multiply by *n*:

The radius of a regular polygon is the distance from the center to any vertex.
In the figure below the leg of the isosceles triangle is a radius *r* of the polygon.
We add a perpendicular *h* from the apex to the base. In this case, let *t* be the whole angle at the apex.
From the figure we see that
The area of any triangle is half the base times height, and since x is already half the base:
Substituting x,h:
Using the double angle 1 trig identity we get
The whole angle *2t* at the apex is dependent on the number of sides: *n*
Substituting this for *2t*:
Finally, there are *n* triangles in the polygon so

The apothem of a regular polygon is a line from the center to the midpoint of a side, which it meets at right angles.
In the figure below, the apothem is labelled *a*.
The whole angle at the apex is dependent on the number of sides *n*:
So the angle *t* is half that, which simplifies to
We know that the tan of an angle is opposite side over adjacent side, so
Transposing we solve this for s
The area of any triangle is half the base times height, so
Which simplifies to
Finally since have *n* triangles, multiply by *n*:

- Polygon general definition
- Quadrilateral
- Regular polygon
- Irregular polygon
- Convex polygons
- Concave polygons
- Polygon diagonals
- Polygon triangles
- Apothem of a regular polygon
- Polygon center
- Radius of a regular polygon
- Incircle of a regular polygon
- Incenter of a regular polygon
- Circumcircle of a polygon
- Parallelogram inscribed in a quadrilateral

- Square
- Diagonals of a square
- Rectangle
- Diagonals of a rectangle
- Golden rectangle
- Parallelogram
- Rhombus
- Trapezoid
- Trapezoid median
- Trapezium
- Kite
- Inscribed (cyclic) quadrilateral

- Regular polygon area
- Irregular polygon area
- Rhombus area
- Kite area
- Rectangle area
- Area of a square
- Trapezoid area
- Parallelogram area

- Perimeter of a polygon (regular and irregular)
- Perimeter of a triangle
- Perimeter of a rectangle
- Perimeter of a square
- Perimeter of a parallelogram
- Perimeter of a rhombus
- Perimeter of a trapezoid
- Perimeter of a kite

- Exterior angles of a polygon
- Interior angles of a polygon
- Relationship of interior/exterior angles
- Polygon central angle

- Tetragon, 4 sides
- Pentagon, 5 sides
- Hexagon, 6 sides
- Heptagon, 7 sides
- Octagon, 8 sides
- Nonagon Enneagon, 9 sides
- Decagon, 10 sides
- Undecagon, 11 sides
- Dodecagon, 12 sides

(C) 2011 Copyright Math Open Reference.

All rights reserved

All rights reserved