Question

Find the area of a regular octagon with an apothem of 8.5.?

Answer 1

`A = 238 " "units^2`

Explanation:

Area of a regular polygon `= 1/2 xx perimeter xx apothem`

`A = (Pa)/2`

Perimeter `= "number of sides" xx "side length"`

`P = ns = 8s`

`A = (8s*8.5)/2 = 34s`

Since we know that the octagon is regular and we know the apothem (height from a side to the center of the polygon, we can calculate the side length):

Each angle of the central angle: `(360^@)/8 = 45^@`

and each side of the polygon forms an isoceles triangle when joined to center. Further, a perpendicular from center to the side of the polygon divides the side in two equal parts and also cuts this angle at the center (`45^@`) in half i.e. `22.5^@`.

Using trigonometry, we can find half the side length (`x`):

`tan 22.5^@ = x/a = x/8.5`

`x = 8.5*tan 22.5^@ ~~ 3.52`

side length: `s = 2x ~~ 7`

`A = 34 * s ~~ 34 * 7 ~~238 " units"^2`