Polygon Angle Calculator

How the Polygon Angle Calculator Works

A regular polygon is a closed, two-dimensional shape in which all sides are equal in length and all interior angles are equal in measure. The polygon angle calculator applies four classical formulas to derive every meaningful angle measurement from a single input: the number of sides (n). These geometric relationships are grounded in Euclidean geometry and are fully documented by the Department of Mathematics at UTSA as part of foundational angle theory.

The Four Core Formulas

Sum of Interior Angles

The combined measure of all interior angles in any simple polygon with n sides is given by:

Sum = (n − 2) × 180°

This result follows from dividing any polygon into triangles by drawing diagonals from one vertex. A polygon with n sides always decomposes into exactly n − 2 non-overlapping triangles, and each triangle contributes 180°. Examples: a quadrilateral (n = 4) has a sum of 360°; a hexagon (n = 6) has a sum of 720°; a decagon (n = 10) has a sum of 1,440°.

Interior Angle of a Regular Polygon

Because all interior angles of a regular polygon are identical, each individual angle equals the total sum divided by the count of vertices:

Interior Angle = [(n − 2) × 180°] ÷ n

• Equilateral Triangle (n = 3): 60°
• Square (n = 4): 90°
• Regular Pentagon (n = 5): 108°
• Regular Hexagon (n = 6): 120°
• Regular Octagon (n = 8): 135°
• Regular Dodecagon (n = 12): 150°

As n grows toward infinity, the interior angle approaches 180° and the polygon approaches a circle. A 1,000-sided polygon has an interior angle of approximately 179.64°.

Exterior Angle

The exterior angle at any vertex is the angle between one side of the polygon and the outward extension of the adjacent side. For regular polygons, this simplifies elegantly:

Exterior Angle = 360° ÷ n

This formula reflects the geometric principle that traversing all the way around any convex polygon — turning at each vertex — completes exactly one full rotation of 360°. For a regular pentagon: 360° ÷ 5 = 72°. At any vertex, the interior angle and exterior angle are supplementary: they always sum to 180°, confirming the two formulas are consistent with each other.

Central Angle

The central angle is the angle formed at the geometric center of the polygon between two radii drawn to adjacent vertices. When a regular polygon is inscribed in a circle, the center divides the full rotation into n equal sectors:

Central Angle = 360° ÷ n

Notably, the central angle and the exterior angle are always equal for every regular polygon. For a regular hexagon: 360° ÷ 6 = 60°. This equivalence explains why a regular hexagon tiles a flat plane using its own circumradius as the inter-center distance.

Variable Reference

• n — Number of sides; must be a positive integer of 3 or greater (a polygon cannot have fewer than 3 sides)
• Interior Angle — Angle formed inside the polygon at each vertex, measured in degrees
• Exterior Angle — Angle between one side and the outward extension of the adjacent side, measured in degrees
• Sum of Interior Angles — Total degree measure of all interior angles combined
• Central Angle — Angle at the polygon center between two adjacent vertex-radii, measured in degrees

Real-World Applications

Polygon angle calculations are fundamental across architecture, engineering, design, and manufacturing. Honeycomb structures in aerospace and packaging exploit the hexagonal 120° interior angle because it maximizes enclosed area per unit of material. Stop signs are regular octagons with 135° interior angles, selected for omnidirectional recognition at road intersections. Tile installers use the angle properties of equilateral triangles (60°), squares (90°), and hexagons (120°) because these are the only three regular polygons whose interior angles divide evenly into 360°, enabling gapless tessellation. Gear teeth in mechanical engineering are profiled based on polygon angle geometry to ensure smooth torque transfer.

As outlined in the algebraic geometry curriculum published by MSTE at the University of Illinois, polygon angle relationships are a cornerstone of applied mathematics with direct relevance to structural analysis, computer graphics, and spatial reasoning. Architects use regular polygon angle properties when designing rotational symmetry in facades, floor plans, and decorative elements.

Quick Reference Table

• Triangle (3): Interior 60° — Exterior 120° — Sum 180°
• Square (4): Interior 90° — Exterior 90° — Sum 360°
• Pentagon (5): Interior 108° — Exterior 72° — Sum 540°
• Hexagon (6): Interior 120° — Exterior 60° — Sum 720°
• Heptagon (7): Interior ≈128.57° — Exterior ≈51.43° — Sum 900°
• Octagon (8): Interior 135° — Exterior 45° — Sum 1,080°
• Nonagon (9): Interior 140° — Exterior 40° — Sum 1,260°
• Decagon (10): Interior 144° — Exterior 36° — Sum 1,440°