Question

I’m struggling with this question , geometry

Answer 1

The exterior angles of a regular n-gon add up to 360º; here n = 9, so each exterior angle has measure (360/9)º = 40º.

Corresponding interior angles are supplementary to exterior angles (they add to 180º), so each interior angle has measure (180 - 40)º = 140º.

Bisect this angle by connecting any given vertex to the center of the nonagon. These connections split up the nonagon into 9 congruent isosceles triangles, each of which have base angles of 70º.

Split each triangle in half to make a total of 18 70º-20º-90º triangles with height h and base b. Then

sin70º = h/7  ==>  h = 6.578

sin20º = b/7  ==>  b = 2.394

and each right triangle has an area of approximately 1/2 b h = 7.874.

Multiply this by 18 to get the area of the nonagon: 141.7.