• Any polygon with n sides can be divided into (n − 2) triangles by drawing the diagonals from one vertex.
• Since the angles of each triangle add up to 180°:

Sum of the interior angles = (n − 2) × 180°

• In a regular polygon each interior angle = (n − 2) × 180° ÷ n.
• The sum of the exterior angles of any polygon is always 360°. In a regular polygon each exterior angle = 360° ÷ n.
• The number of diagonals of a polygon = n (n − 3) ÷ 2. A diagonal is a segment joining two non-adjacent vertices.

Checking the rules on familiar shapes

• For a triangle (n = 3), the sum of the angles is (3 − 2) × 180 = 180°, which is correct.
• For a square (n = 4), the sum is (4 − 2) × 180 = 360°, which is correct because a diagonal divides the square into two triangles.
• The number of diagonals of a triangle is 3 × 0 ÷ 2 = 0, because all three vertices are adjacent.
• The number of diagonals of a square is 4 × 1 ÷ 2 = 2, which are the two well-known diagonals.

• Notice that each interior angle of a regular polygon increases as the number of sides increases, while each exterior angle decreases.