Question

A regular decagon is a polygon with ten sides of equal length. Use complete sentences to design a strategy to determine the sum of the interior angles of a regular decagon. In your final answer, include the sum and all of your calculations.


WRITER

Answer 1

Strategy: Choose one vertex of the decagon and draw diagonals from that vertex to all nonadjacent vertices. This divides the decagon into (n − 2) triangles, where n is the number of sides. Because each triangle has an angle sum of 180 degrees, the sum of the interior angles of the polygon is (n − 2) × 180 degrees.

Calculation using the triangle-decomposition strategy: For a decagon, n = 10, so the number of triangles is 10 − 2 = 8. The sum of the interior angles is therefore 8 × 180 degrees = 1440 degrees. (8 × 180 = 8 × (100 + 80) = 800 + 640 = 1440.)

Check using exterior angles: The sum of the exterior angles of any convex polygon is 360 degrees. For a regular decagon each exterior angle is 360 ÷ 10 = 36 degrees, so each interior angle is 180 − 36 = 144 degrees. Multiplying by 10 gives 10 × 144 = 1440 degrees, confirming the result.

Final answer: The sum of the interior angles of a regular decagon is 1440 degrees.