Polygon Interior Angle Sum Calculator

Definition

An interior angle of a polygon is the angle between two adjacent sides whose interior lies inside the figure. For a convex n-gon (integer n ≥ 3) in the Euclidean plane, the sum of all interior angles is S = (n − 2)·180° = (n − 2)π rad. The symbol n is both the number of sides and the number of vertices. The symbol S is the primary result on this calculator. The number of triangles in a triangulation from one vertex is n − 2; in a regular n-gon one interior angle equals S/n.

The model is a simple convex polygon: every diagonal from a fixed vertex lies inside the figure, and the interiors of the triangulation triangles do not overlap. Algebraically (n − 2)·180° can be evaluated at n = 2 to give 0°, but Euclidean geometry has no 2-gon; the calculator requires n ≥ 3. Concave or star polygons are outside this school-level interior-sum model and are not computed here.

The radian form S = (n − 2)π follows from 180° = π rad. The result panel also shows S in radians, the triangle count n − 2, the regular angle S/n, and the fact that the exterior angles (one at each vertex, consistent turn direction) always sum to 360° = 2π rad, independent of n.

Historical and conceptual context

The fact that the angles of a triangle sum to 180° (or two right angles) belongs to the classical canon of Euclidean geometry and appears in the tradition of Euclid’s Elements as the base for further polygon theorems. Extending the fact to an n-gon by triangulation — splitting the polygon into n − 2 triangles by diagonals from one vertex — is the standard school and textbook derivation. MathWorld (Interior Angle) states the sum (n − 2)π in radians and the regular angle (n − 2)π/n; CRC Standard Mathematical Tables (Zwillinger) collects the same identities among polygon formulas.

Conceptually S measures the total interior turning of a convex n-gon. At n = 3 one recovers the triangle sum; each increase of n by 1 adds exactly one triangle and exactly 180° to the sum. Hence S is linear in n: S(n) = 180°·n − 360°. The coefficient 180° is one triangle; the constant −360° is the expansion of (n − 2)·180°.

In school and engineering practice the formula checks a regular n-gon drawing (angle S/n), supports polygonal mesh reasoning, and converts between degrees and radians. This calculator does not compute areas, perimeters, or the global diagonal count — those live on other pages — but with S and n − 2 it supplies the full angular skeleton of the interior sum.

Formal properties

Input domain: n ∈ ℕ, n ≥ 3 (integer). Output range in degrees: S ∈ {180°, 360°, 540°, …}. In radians: S ∈ {π, 2π, 3π, …}. The map n ↦ S(n) is strictly increasing: S(n + 1) − S(n) = 180°. For a regular polygon the interior angle α = S/n = ((n − 2)/n)·180° satisfies 0° < α < 180° for n ≥ 3; as n → ∞, α → 180°.

Identities: S_deg = (n − 2)·180; S_rad = (n − 2)·π; S_rad = S_deg · π/180; regular α = S/n; triangle count T = n − 2; diagonals from one vertex in the triangulation D_v = n − 3. The exterior-angle sum (one exterior angle per vertex, same orientation) is Σ_ext = 360° for every n. At each vertex of a convex polygon the interior and exterior angles in this convention are supplementary: α + ε = 180°.

Monotonicity and discreteness: n cannot be a fraction. For n = 3, 4, 5, 6 the sums are 180°, 360°, 540°, 720°. A regular hexagon has α = 120°; a square has α = 90°; an equilateral triangle has α = 60°.

Full derivation

Step 1 — triangle. In the Euclidean plane the interior angles of any triangle sum to 180° (π rad). That fact is the starting point.

Step 2 — choose a vertex. Let P be a convex n-gon with vertices v₁, …, vₙ in cyclic order. Fix vertex v₁. The neighbors of v₁ are v₂ and vₙ.

Step 3 — diagonals. Draw the segments v₁v₃, …, v₁vₙ₋₁. There are n − 3 of them. Convexity keeps every such segment inside P.

Step 4 — triangulation and sum. Those diagonals split P into n − 2 triangles. Each contributes 180°, hence S = (n − 2)·180°. In radians: S = (n − 2)π. If P is regular, each interior angle equals S/n.

Special cases

n = 3: S = 180°, T = 1, regular α = 60°. n = 4: S = 360°, T = 2, α = 90°. n = 5: S = 540°, α = 108°. n = 6: S = 720° = 4π rad, α = 120°. n = 8: S = 1080°, α = 135°. n = 12: S = 1800°, α = 150°.

Large-n limit: α → 180°, while S grows like 180°·n. The exterior-angle sum stays 360°.

The case n < 3 is rejected. A non-integer such as n = 4.5 fails the integrality check. The diagram draws triangulation up to n = 16; for larger n the sum S still comes from the formula.

Numerical and computational notes

The calculator uses the single number n. The calculator checks that n is finite, an integer (within 10⁻¹² of rounding), and at least 3. Then S_deg = (n − 2)·180 and S_rad = (n − 2)·π. The primary result is in degrees.

Multiplying the integer (n − 2) by 180 is exact in floating-point arithmetic for ordinary n. The product (n − 2)·π inherits π’s approximation. Comma and period are equivalent.

Related concepts and authorities

The total number of diagonals of an n-gon is n(n − 3)/2 on a separate page — here only the n − 3 diagonals from one vertex appear. The third angle in a triangle (180° − α − β) is the n = 3 case with two known angles.

Authorities: Euclid (triangle sum), MathWorld — Interior Angle, Wikipedia (Interior angle), CRC / Zwillinger. The calculator realizes S = (n − 2)·180°.