Interior angle formula:
(sides − 2) × 180 ÷ sides

Interior angle formula for a regular polygon

Each interior angle of a regular polygon equals (sides − 2) × 180 divided by the number of sides. A regular hexagon has 6 sides, so each angle is 4 × 180 ÷ 6 = 120 degrees.

The (sides − 2) × 180 part is the total of all interior angles, because any polygon can be split into sides minus 2 triangles from one corner, and each triangle adds up to 180 degrees. A regular polygon shares that total equally between its corners.

Worked example

Not every shape comes out as a whole number. A 100-sided polygon has an interior angle of 176.4 degrees, which is nearly a straight line, and the angle approaches 180 degrees as the sides increase.

Going the other way is handy too. If a drawing shows corners of 150 degrees, rearrange the idea: the exterior angle is 30 degrees, so the shape has 360 ÷ 30 = 12 sides. A whole-number answer confirms that a regular polygon with that corner exists, while a fractional one means it does not.

Interior angles by number of sides

Exterior angles, and which shapes tile a floor

Builders and designers meet these angles often. A hexagonal paving slab has 120 degree corners, and a stop sign is a regular octagon with 135 degree corners. To frame a regular octagon in timber, each piece is mitred at half the interior angle, 67.5 degrees, so that two neighbouring pieces meet at the full 135 degrees. The same logic gives a 60 degree cut for a hexagonal planter box.

The exterior angle is 180 minus the interior angle, and for any regular polygon it equals 360 ÷ sides. That gives a second route: for 9 sides, 360 ÷ 9 = 40 degrees, and 180 − 40 = 140 degrees.

The angle matters for tiling. Corners meeting at a point must add to 360 degrees. Squares (4 × 90), equilateral triangles (6 × 60) and regular hexagons (3 × 120) fit with no gaps, which is why floor tiles, honeycombs and paving come in those shapes. Regular pentagons (108 degrees) leave a gap, because 360 does not divide by 108. Regular octagons also leave gaps on their own, which is why octagonal floor tiles are usually paired with small square tiles.

Common mistakes with interior angles

• Forgetting the minus 2. Using sides × 180 gives a total that is 360 degrees too large.
• Applying the formula to an irregular polygon. The shortcut shares the total equally, so it only gives each angle when all angles are equal.
• Mixing up interior and exterior angles. The two on the same corner always add to 180 degrees.
• Entering fewer than 3 sides. A polygon needs at least three.