Interior and Exterior Angles of Polygons

Grade 4

The 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.
Nova
Try this one

The diagram shows quadrilateral PQRS. The interior angles are as shown.

PQRS\(2y°\)\(3y°\)\((y+19)°\)\((y+40)°\)Not drawn accurately

(a) Show that \(y = 43\). [3]

(b) Hence find the size of angle \(PQR\). [1]

Nova's hint:

This question uses the angle sum of a quadrilateral.

Ask yourself: 'What do the four interior angles of any quadrilateral add up to?' Once you have that total, form an equation by adding all four expressions and solve for \(y\).

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