03 // Bidirectional STEM Calculators // MATHθ (°) = ((n - 2) × 180) / n

Regular Polygon Interior Angle & Sum Calculator

Any simple n-sided polygon can be triangulated into (n - 2) triangles totaling (n - 2) × 180°, so each vertex of a regular n-gon measures θ = ((n - 2) × 180) / n degrees.

θ (°) = ((n - 2) × 180) / n

Pick which value you want to calculate (or type into any field):Solving for: θ (°) — Single Interior Vertex Angle

Single Interior Vertex AngleCalculated

Interior angle at each corner of a regular n-gon in degrees

60179.5

deg

Number of Polygon Sides

Vertex / edge count of the regular polygon (n ≥ 3)

3360

unit

Calculated Result · Single Interior Vertex Angle (θ (°))

120

Step-by-Step Calculation

01Canonical Law: θ (°) = ((n - 2) × 180) / n

02Given Inputs (Base Units): n = 6

03Solve for θ (°) (Single Interior Vertex Angle): θ (°) = 120

Any simple n-sided polygon can be triangulated into (n - 2) triangles totaling (n - 2) × 180°, so each vertex of a regular n-gon measures θ = ((n - 2) × 180) / n degrees.

Real-World Reference Values

Equilateral Triangle (n = 3) / Square (n = 4)θ = 60° / θ = 90°

Regular Euclidean tessellation polygons

Regular Pentagon (n = 5) / Hexagon (n = 6)θ = 108° / θ = 120°

Hexagons tile the plane cleanly (3 × 120° = 360°)

Regular Octagon (n = 8) / Decagon (n = 10)θ = 135° / θ = 144°

Exterior angle is always 360° / n

20-Row Comparison Table

Single Interior Vertex Angle (θ (°)) · Number of Polygon Sides (n)

Regular Polygon Interior Angle & Sum Calculator — Number of Polygon Sides → Single Interior Vertex Angle
#	Number of Polygon Sides (n)	Single Interior Vertex Angle (θ (°))	Δ %
01	3	60	-50.00%
02	6 Current	120	0.00%
03	8.4829	137.5617	+14.63%
04	11.2243	147.9269	+23.27%
05	13.9658	154.2227	+28.52%
06	16.7072	158.4525	+32.04%
07	19.4487	161.4898	+34.57%
08	22.1901	163.7766	+36.48%
09	24.9316	165.5605	+37.97%
10	27.673	166.9909	+39.16%
11	30.4145	168.1635	+40.14%
12	33.1559	169.1422	+40.95%
13	35.8974	169.9714	+41.64%
14	38.6388	170.6829	+42.24%
15	41.3803	171.3002	+42.75%
16	44.1217	171.8408	+43.20%
17	46.8632	172.3181	+43.60%
18	49.6046	172.7426	+43.95%
19	52.3461	173.1227	+44.27%
20	55.0875	173.4649	+44.55%

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Frequently Asked Questions

How do I find the number of sides n from a known interior angle?+

Click SOLVE FOR on n (Number of Polygon Sides). Because each exterior angle is (180° - θ) and exterior angles sum to 360°, n = 360 / (180 - θ).

What is the sum of all interior angles in an n-gon?+

Multiply the single vertex angle θ by n, or compute (n - 2) × 180° directly (e.g., 720° for a hexagon).

How many diagonals does an n-sided polygon have?+

A polygon with n vertices has n(n - 3) / 2 diagonals (e.g., 9 diagonals in a hexagon).