Apothem in a regular polygon is the distance from the center of the polygon to the midpoint of a side. In a regular polygon, all the sides are of equal length. Apothem is used to calculate various measures such as the side length, perimeter or the area of a regular polygon. Trigonometric functions are used in order to evaluate the length of the apothem.
Example 1: Find the length of the apothem in a regular hexagon if the side length is 10m.
A regular hexagon has 6 sides of equal length.
Hence it can have 6 equilateral triangles with each angle of
the triangle equal to = 60°
Apothem in the regular hexagon as shown is = AD
The apothem of the polygon divides the angle into half = 30°
Now the entire side length, BC = 10m and DC = 5m
In triangle ADC, tan30° = DC/AD = 5/AD
Apothem, AD = 5/tan30° = 8.66m
Example 2: In the above given regular hexagon, find the area of the hexagon using the length of the apothem.
Area of one triangle in a regular hexagon =1/2 * base * apothem
Given above, base length = side length, BC = 10m
Apothem calculated above, AD = 8.66m
Hence area of the triangle, ABC = 1/2 * BC * AD = 1/2 * 10 * 8.66
So are of the triangle, ABC = 43.3m2
There are total 6 equal triangles in the given hexagon.
Hence area of the regular hexagon = 6 * 43.3 = 259.8m2