Apothem Formula

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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


 

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