SOLUTION: what is the area of a regular nonagon with an apothem of 8 m

Question 1201660: what is the area of a regular nonagon with an apothem of 8 m
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Answer: 209.646864 square meters (approximate)

Explanation:

A regular polygon has equal angles and equal sides.
A nonagon has 9 sides.

Let's draw out a regular nonagon.

If we connected the center to each vertex, then we get 9 congruent mirror copies of the same isosceles triangle.
Think of them as slices of pizza.

Divide 360 degrees over 9 equal pieces.
360/9 = 40

The vertex angle of each isosceles triangle is 40 degrees.

Draw a vertical line through that triangle at the bottom.
This splits the 40 degree angle into two 20 degree pieces.

The apothem is represented as the dashed line.
It is perpendicular to the polygon's edge.

Focus on half of the triangle.

We'll use the tangent ratio to determine x.

tan(angle) = opposite/adjacent
tan(20) = x/8
x = 8*tan(20)
x = 2.911762 approximately
Your calculator needs to be in degree mode.

Double this value to determine the approximate side length of this regular nonagon.
2x = 2*2.911762 = 5.823524

area of one triangle = 0.5*base*height
area of one triangle = 0.5*5.823524*8
area of one triangle = 23.294096

area of nonagon = area of nine triangles
area of nonagon = 9*(area of one triangle)
area of nonagon = 9*(23.294096)
area of nonagon = 9*(23.294096)
area of nonagon = 209.646864 square meters

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