Polygons (Finding Area)

Question

Find the area of a regular octagon with a radius 4 centimeters.

Area = 1/2 (Apothem(Perimeter))

Area= 1/2 (4(Perimeter)

I think I am erroneuosly leaning on a formula that does not seem to work unless you are given specifics regarding side length as well. I am also guessing there is some trig function(a subject I have yet to deep dive into) that needs to be applied?

Thanks for any help,

Brian

Answer 1

draw a diagram

draw the apothem

find half of one side, call it n

when you draw the apothem you form two congruent right triangles

the central angles in each of the eight congruent triangles is 360/8=45°

the apothem bisects this angle so you get 22.5°

use the formula A=ns^2/4tanθ where θ=22.5º, n=8 sides, s=side length

how do we find the side length ?

sin=opposite/hypotenuse

sin22.5º=(1/2)s/4

4sin22.5º=1/2s

8sin22.5º=s

8*0.38268=s

s=3.06 cm

A=8*(3.06)^2/4tan22.5º

A=74.9088/4(0.4142)

A=74.9088/1.6568

A=45.213 cm

if you find the apothem....

cos22.5º=a/4

a=4*0.9238

a=3.6952

Perimeter=8*3.06

P=24.48

A=(1/2)apothem*Perimeter

A=(1/2)(3.6952)(24.48)

A=(1/2)(90.45849)

A=45.23 cm (the slight difference is due to rounding off )

Answer 2

So once again, this is much more easily explained when there's a photo (so pretend the hexagon is actually an octogon).

Since we are trying to find the area this time, we need to first calculate the apothem and the perimeter so that we can plug those values into the formula.

We can find the value of each central angle (or x) by dividing 360/8 = 45°. If each central angle is 45°, then half of that is what we need to use the right triangle at the bottom, so the top angle of the right triangle is 22.5°. You can use trig to find the value of "a" (the apothem, and in this case, the adjacent side) and then use trig again to find the opposite side (which will be half of "s"). Once you find the opposite side, don't forget to double it, and then multiply it by 8 to get the perimeter.

Once you find the apothem and perimeter, you simply plug those values into the formula for area. Do you need help calculating the apothem and opposite side?