Question

The apothem of a decagon is `15`m. Its area is `1595` squared meters. Find the length of one of the sides? Round to `2` decimal places.

Answer 1

Each side is (approximately) 21.27 m.

Explanation:

The apothem of any regular polygon (like a decagon) is like a radius. It is a line from the centre of the polygon to the centre of any of the sides. As such, it is perpendicular to the side it meets, and it is the shortest distance possible from the centre to the perimeter.

The area of a regular decagon has a formula, but it is rarely introduced in geometry units (I had to look it up). But since the decagon can be decomposed into 10 "pizza slices", each of which has its own area of `A_triangle = 1/2 bh`, the whole decagon has area:

`A_"dec" = 10 xx A_triangle`

`color(white)(A_"dec") =10 xx 1/2 bh`

And since we know this height `h` is the apothem (15 m), and the area of the decagon (1595 m²), we can plug these into the formula to get

`1595 " m"^2 =10 xx 1/2 b xx "15 m"`

which allows us to solve for the base `b` (a.k.a. the length of one side of the regular decagon).

`1595 " m"^2 =75b " m"`

`(1595 " m"^2)/(75 " m")= b`

`21.27" m"~~b`

So the length of each side of the regular decagon is (approximately) 21.27 m.