How do we find the apothem of a regular polygon?

1 Answer
Dec 12, 2016

Apothem of a regular polygon with #n# sides and one side as #a# is #a/2cot(pi/n)#.

Explanation:

Apothem is the line joining the center of a regular polygon to the middle point any of its side. It is also the radius of incircle of the regular polygon.

Assume there are #n# sides of the polygon and each side is #a#. Joining center of the polygon to two ends of same side will form an isosceles triangle whose angle at vertex will be #(2pi)/n# and drawing the perpendicular from vertex to side will form a right angle side (as shown below), with altitude forming apothem and angle at vertex being #pi/n#
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and if apothem is #x#, we have

#x/(a/2)=cot(pi/n)#

and hence apothem is #a/2cot(pi/n)#

Hence, apothem of a regular polygon with #n# sides and one side as #a# is #a/2cot(pi/n)#.