How many sides does a polygon have if the sum of its interior is 720?

Answer 1

Assuming it's a regular polygon, it's a hexagon with six sides.

Let n be the unknown number of sides. Since each interior angle is identical and since they sum to 720 degrees, each angle is 720 / n. Now, imagine the unknown polygon is sepparated into equal triangles with their tips at the center and their bases as each of the sides. The sides of the triangle bisect the interior angle, so each base angle of the triangle is (1/2) * (720/n). The third angle of the triangle is 360/n as the triangles all meet to form a perfect circle and there are n of them (one for each side). The two base angles and the thrid angle must sum to 180 degrees as all triangles must in euclidian geometry, so we end up with the equation of (1/2) * (720/n) + (1/2) * (720/n) + 360/n = 180. Solving for n yields n = 6. This method works with any regular polygon.