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Polygon Interior Angle Sum Calculator

Sum the interior angles of any polygon

How it is calculated

For a polygon with n sides the interior angles add up to (n − 2) × 180°. If the polygon is regular, all angles are equal, so each one is (n − 2) × 180° / n. Both values are shown in degrees with two decimals.

The number of sides must be a whole number of at least 3; a value such as 2 or 5.5 gives no result. The per-angle value is only meaningful for a regular polygon (equal sides and equal angles). For an irregular polygon the sum is still correct, but the angles differ and have to be measured individually. The tool gives interior angles only, not exterior ones, which are 180° minus the interior value.

Worked example

For a regular octagon, n = 8. Sum of interior angles = (8 − 2) × 180 = 6 × 180 = 1,080.00 °. Each angle = 1,080 / 8 = 135.00 °. A heptagon (n = 7) would give 900.00 ° in total and about 128.57 ° each, which shows that the per-angle value is not always a round number.

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Common mistakes

  • Applying the per-angle result to an irregular polygon: only the sum holds in general, the individual angles of an irregular shape differ.
  • Entering the number of triangles (n − 2) instead of n: the field is the number of sides, and 6 would be read as a hexagon, not an octagon.
  • Using the interior angle as a miter angle: for a joint where two boards meet, each board is cut at half of the interior angle (67.5° for an octagon).
  • Confusing interior and exterior angles: the exterior angle at each corner of a regular octagon is 45°, not 135°.

About this calculator

For any simple polygon with n sides, the sum of all interior angles is (n − 2) × 180° — a result that comes from splitting the polygon into (n − 2) triangles from a single vertex, each contributing 180°. For a regular polygon, where every angle is equal, dividing that sum by n gives the measure of each individual interior angle: a hexagon's angles sum to 720° and each one measures 120°, for instance.

Geometry students use this to check answers on angle-sum proofs, while designers and fabricators — laying out hexagonal tiles, octagonal gazebos, or CNC-cut polygonal parts — use the per-angle value to set the correct miter or cut angle at each joint so the pieces close up into a clean shape.

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Comments (1)

  • Editorial team

    Check any result against one rule: the angle in each corner of a regular polygon must stay below 180°, and it creeps toward 180° as the side count grows. A 12-sided shape gives 150°, and a 100-sided one gives 176.4°, which is why a many-sided ring looks almost like a circle. If your number is above 180°, the side count was probably mistyped. Which shapes do you lay out most, hexagons, octagons or something with more sides?

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