Question

Please help, i dont understand an ounce of this

Answer 1

Let x be the length of each side of the nonagon. We then split up the nonagon into 9 congruent, isosceles triangles, each with base = x and height = 12. Then the area of each triangle is 1/2 • x • 12 = 6x, so the total area of the nonagon will be 9 • 6x = 54x.

To find x, we can use some facts from geometry and trigonometry.

• In any polygon, the sum of the measures of the exterior angles is 360°. So each of these exterior angles will measure 360°/9 = 40°.

• Exterior angles are supplementary to the interior angles. So each interior angle will measure 180° - 40° = 140°.

• Each of the 9 triangles are isosceles with base angles measuring half the interior angles of the nonagon, 140°/2 = 70°.

• Cut the triangle in half along the labeled inradius of the nonagon, which has length 12. In the resulting right triangle, we have

tan(70°) = 12 / (x/2)

and solving for x gives

tan(70°) = 24/x

x = 24/tan(70°)

x = 24 cot(70°) ≈ 8.7

Then the total area of the nonagon is

54x = 54 • 24 cot(70°) ≈ 471.7