Polygons: Interior and Exterior Angles Explained Simply

A polygon is a closed plane figure made of straight line segments joined end to end. Triangles, quadrilaterals, pentagons, and hexagons are all polygons. The segments are called sides, the joining points are vertices, and the angles inside are interior angles.

Polygons matter because their angle rules are among the most reliable patterns in mathematics: a few simple formulas describe the angles of every polygon with any number of sides. These rules unlock tessellations, star shapes, and countless exam questions.

Key Concepts

• Regular vs irregular: a regular polygon has all sides and all angles equal (equilateral triangle, square, regular pentagon); an irregular polygon does not.
• Interior angle: the angle inside the polygon at a vertex. Exterior angle: the angle formed by one side and the extension of the adjacent side; at every vertex, interior + exterior = 180°.
• Sum of exterior angles: for any convex polygon, the exterior angles (one per vertex) always add to 360°, whether it has 3 sides or 300.
• Triangulation idea: an n-sided polygon can be split into (n − 2) triangles by drawing diagonals from one vertex, which is why the interior sum formula works.
• Convex polygon: all interior angles are less than 180° and the whole figure bulges outward; the standard formulas assume convex polygons.

Formulas

• Sum of interior angles: (n − 2) × 180°, where n is the number of sides. For a hexagon: (6 − 2) × 180° = 720°.
• Each interior angle of a regular polygon: (n − 2) × 180° / n. For a regular octagon: 6 × 180° / 8 = 135°.
• Each exterior angle of a regular polygon: 360° / n. This is often the fastest route to the interior angle via 180° − exterior.
• Number of diagonals: n(n − 3) / 2. A pentagon has 5 × 2 / 2 = 5 diagonals.

Worked Example

Find each interior and exterior angle of a regular pentagon.

1. Use the exterior angle first: 360° / 5 = 72°.
2. Interior angle: 180° − 72° = 108°.
3. Verify with the interior formula: (5 − 2) × 180° / 5 = 3 × 180° / 5 = 540° / 5 = 108°. Matches.
4. Verify the sum: 5 × 108° = 540° = (5 − 2) × 180°. Correct.

Common Misconceptions

• The exterior angle sum is always 360°. Students often think it grows with the number of sides, but it is constant for every convex polygon.
• Interior and exterior are different angles. The 108° of a regular pentagon is the interior angle; the exterior angle is 72°. Mixing them up ruins the calculation.
• The (n − 2) × 180° rule gives the sum, not one angle. For a regular polygon you must divide by n to get each angle.

Real-World Applications

• Nature: honeycomb cells are regular hexagons because 120° interior angles tile the plane with minimal wax.
• Traffic signs: the stop sign is a regular octagon, chosen because its distinctive 8-sided shape is recognizable even when covered in snow.
• Architecture: domes and floor tilings use regular pentagons, hexagons, and octagons for strength and beauty.
• Games: a football (soccer ball) pattern combines pentagons and hexagons, and dice are built from regular polygons.

Every n-sided polygon has interior angles summing to (n − 2) × 180°, while its exterior angles always sum to 360°. For regular polygons, dividing by n gives each individual angle. With these two facts you can find any missing polygon angle in seconds — just remember which angle is interior and which is exterior.