Interior and Exterior Angles of Polygons
Grade 4The angle sum formula
For any polygon with n sides, the sum of its interior angles is given by a formula that comes from splitting the polygon into triangles.
(n − 2) × 180°
For a triangle (n = 3): (3 − 2) × 180° = 180°. For a quadrilateral (n = 4): (4 − 2) × 180° = 360°. Each extra side adds another 180°.
Regular polygons
In a regular polygon, all sides are equal and all angles are equal. So each interior angle = total interior angle sum ÷ number of sides.
Each exterior angle of a regular polygon = 360° ÷ n. Interior and exterior angles at any vertex always add up to 180°.
The exterior angles of any polygon (regular or not) always sum to 360° — a full turn. This is a handy shortcut for many questions.
Worked example
EXAMPLE
Find the interior angle of a regular octagon.
An octagon has n = 8 sides. Interior angle sum = (8 − 2) × 180° = 6 × 180° = 1080°. Each interior angle = 1080° ÷ 8 = 135°.
Watch out
The formula (n − 2) × 180° gives the total for all angles. Divide by n only if the polygon is regular. For an irregular polygon you must use the total and the angles you already know.
