Polygon interior angles:
the sum and the size of each angle

Why (n − 2) × 180° works

Pick one corner of any polygon and draw lines from it to every other corner that it does not touch. A four-sided shape is cut into 2 triangles, a five-sided one into 3, a six-sided one into 4. In every case the number of triangles is two fewer than the number of sides.

Each triangle has angles summing to 180°, and together the triangles fill the interior of the polygon exactly. So the interior angles of the polygon add up to the number of triangles times 180°.

Each angle of a regular polygon

A regular polygon has equal sides and equal angles, so you can share the sum equally. Divide the total by the number of corners.

The 120° angle is why hexagons tile a flat surface: three of them fit around a point, 3 × 120 = 360. It is also why honeycombs and bathroom tiles are hexagonal.

A related fact is useful on a workshop floor. To close a frame made of equal pieces, each joint must be cut at half the interior angle. For a regular octagon that is 135 ÷ 2 = 67.5° from the edge, and for a hexagon it is 60°. Cutting at the wrong value leaves a gap at the last corner, which grows with every joint, so it is worth checking with a spare offcut before cutting the whole set.

Remember too that the formula describes interior angles only when the polygon is simple. A star shape drawn with crossing lines, such as a pentagram, follows different rules.

Reference values from triangle to dodecagon

Read down the right-hand column: exterior angles are 360 ÷ n. At each corner the interior and exterior angles sit on a straight line, so they add to 180°. This is a quick way to check any entry.

Working backwards: how many sides?

If a regular polygon's interior angle is given, find the exterior angle first, 180° minus the interior, then divide 360° by it.

If the answer is not a whole number, no regular polygon has that angle. An interior angle of 100°, for instance, gives 360 ÷ 80 = 4.5 sides, which is impossible.

Irregular polygons and the missing angle

For a shape that is not regular, the total is the same but the angles differ. A common exercise is finding the last angle when the others are known.

The rule holds for concave shapes too, where one interior angle exceeds 180°, as long as the outline does not cross itself.

The exterior angle shortcut

Walk around a convex polygon, turning at each corner. By the time you return to the start you have turned through exactly one full circle, 360°. That is why the exterior angles of any convex polygon sum to 360°, whatever the number of sides.

For a regular polygon the turns are equal, so each is 360 ÷ n. The interior angle is then 180° minus that. For a pentagon, 360 ÷ 5 = 72 and 180 − 72 = 108. This is often faster than the main formula, and it explains why the interior angle approaches 180° as n grows: a polygon with many sides looks like a circle.

Another use is a computer drawing. To draw a regular n-gon by turtle commands, move forward a fixed length and turn by 360 ÷ n degrees, n times.

Diagonals, triangles and sanity checks

The number of triangles in the fan from one vertex is n − 2, and the number of diagonals from one vertex is n − 3. A hexagon has three diagonals from each corner, producing four triangles. The count is a handy check that you have used the right n.

• A seven-sided shape has a total of (7 − 2) × 180 = 900°, but a regular heptagon's angle, 128.57°, does not round to a whole number.
• A polygon cannot have fewer than three sides, and n = 3 gives the familiar 180°.
• If the angles you have measured on a drawing do not add to the expected total, one of them is probably an exterior angle or a reflex angle read the wrong way.

Where this shows up

• Cutting mitres: a regular octagonal frame needs corner cuts of half the interior angle, 67.5° each, or 22.5° off straight.
• Tiling and floors: shapes whose interior angle divides 360° evenly (3, 4, 6 sides) tile a plane on their own.
• Garden and fence design: a regular pentagon bed has 108° corners, which affects how rails meet.
• Turtle graphics and robot paths: the turn at each corner is the exterior angle, 360 ÷ n.