Question

How can I find the area of an 18gon with a radius length 1ft

Answer 1

Answer:

Step-by-step explanation:

If we draw radii from the center of this 18-gon to each vertex, we end up with 18 triangles that are all congruent to each other. If we divide 360 by 18 we get the measure of the vertex angle of each of these 18 triangles. 360/18 = 20. So we take one of these triangles and pull it out and use it as a representative triangle from which we use trig to find its height and its base length.

The formula for the area for a regular polygon is

$A=\frac{1}{2}ap$ where a is the apothem (the height of our triangle) and p is the perimeter (found from the base of our triangle).

If we split this triangle we extracted right down the center vertically, we get 2 right triangles. Pull one of those out and this is the triangle we work with. The vertex angle is split in half and it now measures 10. The one base angle is the right angle and, by the Triangle Angle-Sum Theorem, the other base angle is 80. To find the height (or the apothem) of this right triangle that has a hypotenuse of 1 (the radius of the polygon is the hypotenuse of the right triangle), we use the sin or cos ratio. I used cos:

$cos10=\frac{a}{1}$ so

a = .9848

To find the base of this right triangle, I used the sin ratio:

$sin10=\frac{p}{1}$ so

p = .1736. But this is only half the length of one of the sides of the polygon, so multiply it by 2 to get the length of the whole side.

.1736 * 2 = .3473

Now, in order to find the perimeter, we multiply that one side by 18 to get

p = 6.2514

Now we're ready to find the area:

$A=\frac{1}{2}(.9848)(6.2514)$ so

A = 3.078 square feet