Question

If apothem of a polygon of `18` sides is `15.5` then what is its area and perimeter?

Answer 1

Area of polygon is `762.526` and perimeter is `49.195`.

Explanation:

In a regular polygon, apothem is the length of the line segment from the center of the regular polygon to the midpoint of a side.

As we have a polygon with `18` sides, each side subtends an angle of `20^@` at the center forming an isosceles triangle and area of polygon will be `18` times the area of this polygon. Now consider the following figure.

Let `h` be the apothem, then `MB=hxxtan10^@` and `AB=2hxxtan10^@` and area of isosceles triangle is

`(2hxxtan10^@xxh)/2=h^2tan10^@` and area of polygon is

`18h^2tan10^@` and as `h=15.5`

Area of polygon is `18xx15.5^2xxtan10^@`

= `18xx15.5^2xx0.176327=762.526`

and perimeter is `18AB=18xx15.5xx0.176327=49.195`

Note - we can say that if `a` is the length of the apothem of a polygon of `n` sides, its perimeter is `2natan(pi/n)` and area is `na^2tan(pi/n)`.