Which expression is the sum of the interior angles of a polygon with n sides?

Which expression is the sum of the interior angles of a polygon with n sides?

• A. S=n*180
• B. A=bh/2
• C. S=(n-2)180
• D. P=n+s

Answer: (C)

Think about how polygons can be broken into triangles. If you draw diagonals from one vertex, you can divide an n-sided polygon into exactly n−2 triangles. Each triangle has interior angles that sum to 180 degrees, so the whole polygon’s interior angles add up to (n−2) × 180 degrees. That’s why the expression (n−2)180 matches the sum. For example, a triangle (n=3) gives (3−2)×180 = 180; a quadrilateral (n=4) gives 360; a pentagon (n=5) gives 540. The other expressions don’t fit these facts: one would give 180n, which doesn’t match the actual sums; another is a formula for area, and the last isn’t a correct way to express an angle sum.

Think about how polygons can be broken into triangles. If you draw diagonals from one vertex, you can divide an n-sided polygon into exactly n−2 triangles. Each triangle has interior angles that sum to 180 degrees, so the whole polygon’s interior angles add up to (n−2) × 180 degrees. That’s why the expression (n−2)180 matches the sum.

For example, a triangle (n=3) gives (3−2)×180 = 180; a quadrilateral (n=4) gives 360; a pentagon (n=5) gives 540. The other expressions don’t fit these facts: one would give 180n, which doesn’t match the actual sums; another is a formula for area, and the last isn’t a correct way to express an angle sum.