Polygons (Finding Apothem)

Question

I am trying to find the apothem for a nonagon with sides each equaling 8.

I understand the formula for regular polygon to be : Area= 1/2(apothem)(perimeter)

For my problem the perimeter is 72, though I am not sure how, with this information, I can find the measure of the apothem.

Any clarification would be most helpful.

Thanks,

Brian

Answer 1

apothem = side/[2 * tan(180/n)], where side = 8 and n = 9, so apothem = 4/tan(20) = 10.99

Answer 2

Okay, this is going to be very difficult to explain without a picture, so bare with me.

I realize the photo above is hexagon, but it demonstrates the same general concept.

First you want to think about what x would be if you had 9 sides. You would simply need to take 360 and divide by 9 to get 40. Notice, the apothem would cut that angle in half, so if we're looking at using one of the right triangles drawn to find a, then that top angle of the right triangle would be half of 40 which is 20. Then you can use trigonometry to find the value of the apothem. The reference angle is 20°, the opposite side is 4 (half the length of a side of the polygon), and you're trying to solve for the adjacent side. So you want to use the tangent function. Can you solve from there?