What Is the Interior-Angle Sum?

Where the formula comes from

A common explanation is triangulation. From one vertex of a convex n-gon, you can divide the polygon into n minus two triangles, and each triangle contributes one hundred eighty degrees to the total.

This formula works for convex and concave polygons alike as long as the polygon is simple. It is about the full interior total, not about the size of one single angle unless the polygon is regular.

Choose one vertex and draw diagonals to the non-neighboring vertices. In a convex polygon with n sides, this creates n minus 2 triangles. Since each triangle contributes 180 degrees, the total becomes (n minus 2) times 180 degrees. The drawing supplies the reason for the formula.

• The interior angle sum of an n-gon is (n minus 2) times 180 degrees.
• The interior-angle sum of an n-gon is 180 times (n minus 2) degrees.
• The formula gives the total of all interior angles, not the measure of one interior angle.
• In a regular polygon, each interior angle is found by dividing that total equally among the n angles.

When the interior sum helps

• Use the interior-angle sum formula to find the total interior measure of a polygon from its side count.
• Use it in regular-polygon problems to find one interior angle.
• Use it in reverse when solving for the number of sides from a known total interior sum.

Interior-sum errors to catch

• Do not confuse the interior-angle sum with the measure of one interior angle.
• Do not forget to use n minus 2, not n itself, in the formula.
• Do not apply regular-polygon equal-angle reasoning unless the polygon is stated to be regular.