多角形内角和計算機

The Polygon Angle Sum Calculator returns the total of all interior angles in any polygon, given only the number of sides. The formula is one of the cleanest in plane geometry: (n − 2) × 180°. This same formula, divided by n, gives the measure of each individual interior angle when the polygon is regular (all sides and angles equal). This tutorial proves the formula by triangle decomposition, walks through interior and exterior angles for the most common polygons, and explains why exterior angles always sum to exactly 360° regardless of n.

The interior angle sum formula

For any simple polygon (no self-crossings) with n sides:

Interior angle sum = (n − 2) × 180°

The formula works for both convex and concave polygons. It does not require regularity — irregular polygons with the same number of sides have the same total angle sum, even though their individual angles differ.

Why the formula is (n − 2) × 180°

Pick any vertex of an n-sided polygon. Draw diagonals from that vertex to every other non-adjacent vertex. You will have drawn n − 3 diagonals (one to each of the n − 3 non-adjacent vertices — you cannot draw a diagonal to the two adjacent ones or to yourself).

These n − 3 diagonals divide the polygon into n − 2 triangles. The interior angles of each triangle sum to 180°. Total: (n − 2) × 180°.

This proof works for any convex polygon directly. For concave polygons, you may need to pick the vertex carefully so the diagonals stay inside the figure, but the count of triangles is still n − 2.

Worked tables for common polygons

The "each interior" column only applies if the polygon is regular. An irregular pentagon still has interior sum 540°, but the five angles can be anything summing to 540°.

The exterior angle sum is always 360°

An exterior angle at a vertex is the supplement of the interior angle: exterior = 180° − interior. Equivalently, an exterior angle is what you would turn through if you walked along the boundary and turned at each vertex to follow the next side.

For any convex polygon, the exterior angles sum to exactly 360° — regardless of n. Geometric intuition: if you walk all the way around the polygon, you make exactly one full turn (360°) by the time you return to your starting orientation. The total turn equals the sum of all the individual turns at each vertex.

For a regular polygon, each exterior angle equals 360° / n. So a regular hexagon has exterior angles of 60° (and interior angles of 120°, since 60° + 120° = 180°).

Why interior + exterior at each vertex = 180°

The interior angle and the exterior angle at the same vertex form a linear pair — they are on opposite sides of the same vertex, sharing one side. A linear pair sums to 180°. So:

interior + exterior = 180°

For a regular polygon:

(n − 2) × 180° / n + 360° / n = 180°

You can verify: ((n − 2) × 180° + 360°) / n = (180n − 360 + 360) / n = 180n / n = 180°. ✓

Finding the number of sides from an interior angle

If you know each interior angle of a regular polygon, you can solve for n. From each interior = (n − 2) × 180° / n:

n × (each interior) = (n − 2) × 180°
n × (each interior) = 180n − 360
180n − n × (each interior) = 360
n(180 − each interior) = 360
n = 360 / (180 − each interior)

Example: each interior = 144°. Then n = 360 / (180 − 144) = 360 / 36 = 10. Regular decagon.

Worked examples

Example 1 — sum for n = 7: (7 − 2) × 180° = 5 × 180° = 900°.

Example 2 — each interior for regular n = 12: (12 − 2) × 180° / 12 = 1800° / 12 = 150°.

Example 3 — find n from a given regular interior of 162°: n = 360 / (180 − 162) = 360 / 18 = 20. Regular icosagon.

Real-world applications

• Tiling and tessellation. A polygon can tile the plane (alone, edge-to-edge) only if its interior angle divides 360° evenly. Triangles (60° each), squares (90°), and regular hexagons (120°) are the only regular polygons that can tile the plane alone. Pentagons (108°) cannot — 360°/108° is not an integer.
• Architecture and design. The interior angle of a regular polygon dictates corner cuts in wood, metal, or glass when building n-sided structures (gazebos, planters, picture frames).
• Crystallography and chemistry. Molecular geometries (trigonal, square planar, octahedral, etc.) describe bond angles at the central atom — exactly the interior angles of regular polygons.
• Game design and graphics. Procedural generation of polygons (city plans, asteroids, geodesic domes) relies on (n − 2) × 180° to compute correct angles.

Common mistakes

• Using n × 180° instead of (n − 2) × 180°. The formula subtracts 2 first. Without that, you would be over-counting by 360°.
• Applying the regular-polygon "each interior" formula to irregular polygons. Irregular polygons have the same sum but different individual angles.
• Confusing interior with exterior. Interior is inside the polygon. Exterior is outside, the supplement.
• Using the formula on self-intersecting figures. Star polygons (e.g., pentagrams) do not satisfy (n − 2) × 180° in the standard sense — their "interior angle sums" depend on which crossings are interior.