Question

A regular decagon has a radius of 14'. Determine the length of the apothem. Then determine the perimeter of the decagon

Answer 1

Part 1) The length of the apothem is 13.32'

Part 2) The perimeter of the decagon is 86.5'

Explanation:

we know that

A regular decagon has 10 equal sides and 10 equal interior angles

A regular decagon can be divided into 10 congruent isosceles triangle

(they are isosceles since their two sides are the radii of the polygon and the unknown side is the side of the polygon)

The vertex angle of each isosceles triangle is equal to

`(360^o)/(10)=36^o`

To find out the side length of the decagon, we can use the law of cosines

so

`c^2=a^2+b^2-2(a)(b)cos(C)`

where

c is the length side of decagon

a and b are the radii

we have

`a=14'\\b=14'\\C=36^o`

substitute the values

`c^2=14^2+14^2-2(14)(14)cos(36^o)`

`c^2=392-(392)cos(36^o)`

`c^2=392-(392)cos(36^o)`

`c=8.65'`

To fin out the perimeter of decagon multiply the length side by 10

so

`P=8.65(10)=86.5'`

To find out the apothem we can apply the Pythagorean Theorem in one isosceles triangle

see the attached figure to better understand the problem

`r^2=a^2+(c/2)^2`

substitute the given values

`14^2=a^2+(8.65/2)^2`

solve for a

`a^2=14^2-(8.65/2)^2`

`a=13.32'`