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.
1 Answer
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_"dec" = 10 xx A_triangle#
#color(white)(A_"dec") =10 xx 1/2 bh#
And since we know this height
#1595 " m"^2 =10 xx 1/2 b xx "15 m"#
which allows us to solve for the base
#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.