How do we find the apothem of a regular polygon?

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How do we find the apothem of a regular polygon?

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Answer 1 · 2016-12-12T11:37:25

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 $a/2cot(pi/n)$ .

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How do we find the apothem of a regular polygon?

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