The sum of the interior angles of a polygon with n sides is what formula (in degrees) for simple polygons?

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Multiple Choice

The sum of the interior angles of a polygon with n sides is what formula (in degrees) for simple polygons?

Triangulating the polygon reveals the idea behind the sum. A simple polygon with n sides can be divided into exactly n−2 nonoverlapping triangles by drawing diagonals from one vertex. Since every triangle has interior angles adding up to 180 degrees, the total interior angle sum for the polygon is (n−2)×180 degrees. This matches small cases: a triangle gives 180 degrees, a quadrilateral 360, a pentagon 540, and so on. The other expressions don’t fit because they’d imply more triangles than exist or misstate each triangle’s angle sum. 180n would treat the polygon as if it contained n triangles, which isn’t true; and (n−2)×90 would assume each triangle contributes only 90 degrees, whereas a triangle’s interior angle sum is 180.

Triangulating the polygon reveals the idea behind the sum. A simple polygon with n sides can be divided into exactly n−2 nonoverlapping triangles by drawing diagonals from one vertex. Since every triangle has interior angles adding up to 180 degrees, the total interior angle sum for the polygon is (n−2)×180 degrees. This matches small cases: a triangle gives 180 degrees, a quadrilateral 360, a pentagon 540, and so on.

The other expressions don’t fit because they’d imply more triangles than exist or misstate each triangle’s angle sum. 180n would treat the polygon as if it contained n triangles, which isn’t true; and (n−2)×90 would assume each triangle contributes only 90 degrees, whereas a triangle’s interior angle sum is 180.

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