Interior and Exterior Angles of Polygons: KS3 Maths

Every polygon has interior angles (inside) and exterior angles (outside). The sum of interior angles of any polygon with n sides is (n − 2) × 180°. The exterior angles of any convex polygon always add up to exactly 360°, no matter how many sides it has.

What are interior and exterior angles?

An interior angle is the angle inside the polygon at each vertex. An exterior angle is formed by extending one side of the polygon beyond the vertex — it sits outside the shape.

At every vertex, the interior and exterior angles are on a straight line, so they add up to 180°:

Interior angle + Exterior angle = 180°

This relationship holds for every vertex on any convex polygon.

How do you find the sum of interior angles?

Use the formula: Sum of interior angles = (n − 2) × 180°, where n is the number of sides.

The formula comes from splitting the polygon into triangles from one vertex. A polygon with n sides splits into (n − 2) triangles, and each triangle contains 180°.

Polygon | Sides (n) | (n − 2) × 180° | Sum of interior angles
Triangle | 3 | 1 × 180° | 180°
Quadrilateral | 4 | 2 × 180° | 360°
Pentagon | 5 | 3 × 180° | 540°
Hexagon | 6 | 4 × 180° | 720°
Octagon | 8 | 6 × 180° | 1080°

How do you find one interior angle of a regular polygon?

In a regular polygon all sides are equal and all angles are equal. Divide the angle sum by the number of sides:

One interior angle = (n − 2) × 180° ÷ n

Worked example — regular hexagon:

A hexagon has n = 6 sides.

Sum of interior angles = (6 − 2) × 180° = 4 × 180° = 720°.

Each interior angle = 720° ÷ 6 = 120°.

Why do exterior angles always add up to 360°?

Imagine walking all the way around a convex polygon, turning at each vertex. By the time you return to your starting point you have completed exactly one full rotation — 360°. Each turn equals the exterior angle at that vertex, so the exterior angles sum to 360°.

This fact is true for any convex polygon, not just regular ones.

How do you find one exterior angle of a regular polygon?

For a regular polygon with n sides, all exterior angles are equal:

One exterior angle = 360° ÷ n

Worked example — regular pentagon:

A pentagon has n = 5 sides.

Each exterior angle = 360° ÷ 5 = 72°.

Check with the interior angle: 180° − 72° = 108°. And (5 − 2) × 180° ÷ 5 = 540° ÷ 5 = 108°. ✓

How do you find a missing angle in an irregular polygon?

Add all the known interior angles and subtract from the total sum.

Worked example — irregular pentagon:

A pentagon has four known interior angles: 95°, 110°, 130°, 105°. Find the fifth angle.

Sum of all interior angles of a pentagon = (5 − 2) × 180° = 540°.

Total of known angles = 95° + 110° + 130° + 105° = 440°.

Missing angle = 540° − 440° = 100°.

Check: 95° + 110° + 130° + 105° + 100° = 540°. ✓

What mistakes should you watch out for?

Using the wrong formula. The formula is (n − 2) × 180°, not n × 180°. The minus 2 accounts for the fact that a polygon needs at least three sides.

Confusing interior and exterior angles. The question may give an exterior angle and ask for the interior one — always check which angle you have been given.

Applying the exterior angle formula to irregular polygons. The shortcut 360° ÷ n only gives equal angles for regular polygons. For irregular polygons, use interior angle + exterior angle = 180° at each vertex individually.

Frequently asked questions

What is the formula for the sum of interior angles of a polygon?

The formula is (n − 2) × 180°, where n is the number of sides. It works for any polygon. For a triangle (n = 3) it gives 180°; for a quadrilateral (n = 4) it gives 360°, which you can verify by splitting a rectangle into two triangles.

Do exterior angles always add up to 360°?

Yes — for any convex polygon. As you walk around the perimeter and turn at each corner, you complete exactly one full rotation of 360°. This holds regardless of the number of sides or whether the polygon is regular or irregular.

How do you find the number of sides if you know one interior angle of a regular polygon?

Rearrange the formula: n = 360° ÷ (180° − interior angle). For example, if each interior angle is 135°, then n = 360° ÷ (180° − 135°) = 360° ÷ 45° = 8 sides (a regular octagon).

What is the difference between a regular and irregular polygon?

A regular polygon has all sides equal and all angles equal (e.g. a square, equilateral triangle, regular hexagon). An irregular polygon has sides or angles that are not all equal. The angle sum formulas apply to both, but the shortcut "divide by n" for one angle only works for regular polygons.

For Socratic geometry practice with polygons, see aitutors.me.