多角形辺計算機

The Polygon Sides Calculator answers the reverse of the standard polygon-angle problem: instead of "given n sides, find the interior angle sum", it asks "given the angle sum (or a single regular-polygon interior angle), how many sides does the polygon have?". Both reverses fall out of the same identity — this tutorial derives it from scratch and walks 3 worked examples.

The master identity

For any convex polygon with n sides:

Interior angle sum S = (n − 2) × 180°

Why: pick any vertex of the polygon and draw all diagonals from it to the non-adjacent vertices. This cuts the polygon into exactly (n − 2) non-overlapping triangles (3-gon → 1 triangle, 4-gon → 2, 5-gon → 3, etc.). Each triangle's angles sum to 180°, so the polygon's interior angles sum to (n − 2) × 180°.

The companion identity: exterior angle sum = 360° always, regardless of n. (Each exterior angle = 180° − corresponding interior angle. Summing: n × 180° − (n − 2) × 180° = 2 × 180° = 360°.)

Inverse 1: find n from S (any polygon)

Rearrange S = (n − 2) × 180° for n:

n = S / 180° + 2  (or equivalently n = (S + 360°) / 180°)

Example 1: S = 1080°. n = 1080 / 180 + 2 = 6 + 2 = 8 sides (octagon).

Example 2: S = 3240°. n = 3240 / 180 + 2 = 18 + 2 = 20 sides (icosagon).

This inverse works for any polygon — regular or irregular — because the angle-sum identity holds universally.

Inverse 2: find n from one interior angle (regular polygons only)

For a regular polygon (all sides + angles equal), each interior angle = (n − 2) × 180° / n. Rearrange for n:

n = 360° / (180° − each interior angle)

Derivation: let i = each interior. Then i = (n − 2) × 180° / n → n·i = 180n − 360 → 360 = n(180 − i) → n = 360 / (180 − i).

Example 3: i = 135°. n = 360 / (180 − 135) = 360 / 45 = 8 sides (regular octagon).

Example 4: i = 144°. n = 360 / 36 = 10 sides (regular decagon).

Example 5: i = 60°. n = 360 / 120 = 3 sides (equilateral triangle — the only regular polygon with 60° interior angles).

Sanity checks

• n must be a whole number ≥ 3. If your computation gives n = 4.7 or n = 2.3, the input is invalid: no polygon has 4.7 sides, and "polygons" with < 3 sides don't exist.
• Each interior angle for a convex regular polygon is in (60°, 180°). Below 60° → not enough vertices to close. Equal to 180° → not a polygon (straight line). Approaching 180° → very many sides (e.g. 175° → 72 sides).
• Angle sum scales linearly with n. Each extra side adds exactly 180° to S. Quick mental check: pentagon = 540°, hexagon = 720°, heptagon = 900°.

Common mistakes

• Using "n = S / 180" without the +2. The identity is (n − 2) × 180, so the inverse adds 2. Forgetting the +2 underestimates n by 2.
• Applying the "each interior" formula to irregular polygons. n = 360 / (180 − i) ONLY works if all interior angles are equal (regular polygon). For irregular polygons, you must use the angle-sum inverse with the TOTAL of all interior angles.
• Confusing interior with exterior angles. Each exterior = 180° − each interior. If your input is exterior (e.g. "exterior angle 45°"), first convert: i = 180° − 45° = 135°, then apply the formula.
• Treating concave polygons. The identity assumes a CONVEX polygon (no interior angle > 180°). For concave / self-intersecting cases, decompose into convex pieces first.

When to use a different calculator

• For the forward direction (n → S or n → each interior), use the Polygon Angle Sum Calculator.
• For finding regular polygon area / perimeter / vertex coordinates from n + side length, see the standard polygon area tool in the Polygon hub.
• For computing the interior or exterior angle of irregular polygons from coordinates, the Shoelace + interior-angle approach lives in the coordinate-based calculator.