Answer:
The area of the polygon = 216.4 mm²
Step-by-step explanation:
* Lets talk about the regular polygon
- In the regular polygon all sides are equal in length
- In the regular polygon all interior angles are equal in measures
- When the center of the polygon joining with its vertices, all the
triangle formed are congruent
- The measure of each vertex angle in each triangle is 360°/n ,
where n is the number of its sides
* Lets solve the problem
- The polygon has 9 sides
- We can divide it into 9 isosceles triangles all of them congruent,
if we join its center by all vertices
- The two equal sides in each triangle is 8.65 mm
∵ The measure of the vertex angle of the triangle = 360°/n
∵ n = 9
∴ The measure of the vertex angle = 360/9 = 40°
- We can use the area of the triangle by using the sine rule
∵ Area of the triangle = 1/2 (side) × (side) × sin (the including angle)
∵ Side = 8.65 mm
∵ The including angle is 40°
∴ The area of each triangle = 1/2 (8.65) × (8.65) × sin (40)°
∴ The area of each triangle = 24.04748 mm²
- To find the area of the polygon multiply the area of one triangle
by the number of the triangles
∵ The polygon consists of 9 congruent triangles
- Congruent triangles have equal areas
∵ Area of the 9 triangles are equal
∴ The area of the polygon = 9 × area of one triangle
∵ Area of one triangle = 24.04748 mm²
∴ The area of the polygon = 9 × 24.04748 = 216.42739 mm²
* The area of the polygon = 216.4 mm²
