The regular polygon

The area of the regular polygon is $A = \frac{P e r i m e t e r \cdot a p o t h e m}{2}$

The perimeter is: $P = n \cdot l$

with $l$ being the length of the radius and $n$ the number of sides.

Example

Calculate the area of a hexagon of $l = 10 c m$

Using Pythagoras we find the apothem $a$ , or the height of one of six equilateral triangles that form the hexagon.

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$l^{2} = a^{a} + (\frac{l}{2})^{2} a^{2} = l^{2} \cdot (1 - \frac{1}{4}) a = \frac{\sqrt{3}}{2} \cdot l$

The area of one of the triangles: $A_{t r i a n g l e} = \frac{l \cdot a}{2} = l^{2} \cdot \frac{\sqrt{3}}{4} = 25 \sqrt{3} c m^{2}$
Finally, multiplying by $6$ we obtain the total area: $A_{h e x a g o n} = 150 \sqrt{3} c m^{2}$

Note: To find areas of more complex irregular polygons, the philosophy will be the same: to break them down into triangles and to add the areas of the triangles.