Polygon Angle Calculator
Calculate interior and exterior angles of regular polygons with detailed explanations
Calculate Polygon Angles
Minimum 3 sides required for a polygon
Choose your preferred angle measurement unit
Example Calculations
Regular Hexagon (6 sides)
Interior angle: (6-2)π/6 = 4π/6 = 2π/3 ≈ 2.094 rad
In degrees: 2π/3 × 180°/π = 120°
Exterior angle: 2π/6 = π/3 ≈ 1.047 rad = 60°
Sum of interior angles: (6-2) × 180° = 720°
Regular Octagon (8 sides)
Interior angle: (8-2)π/8 = 6π/8 = 3π/4 ≈ 2.356 rad
In degrees: 3π/4 × 180°/π = 135°
Exterior angle: 2π/8 = π/4 ≈ 0.785 rad = 45°
Sum of interior angles: (8-2) × 180° = 1080°
Common Polygon Angles
Interior angles in degrees
Polygon Facts
Sum of exterior angles is always 360° for any polygon
Interior + Exterior angle = 180° (supplementary)
As sides increase, interior angles approach 180°
Regular polygons have all equal sides and angles
Central angle equals exterior angle in regular polygons
Understanding Polygon Angles
Interior Angles
Interior angles are formed inside the polygon at each vertex where two sides meet. In a regular polygon, all interior angles are equal.
Key Properties
- •Formula: α = (n-2) × 180° / n
- •Sum of all interior angles: (n-2) × 180°
- •Always less than 180° in convex polygons
- •Approaches 180° as sides increase
Exterior Angles
Exterior angles are formed between one side of the polygon and the extension of an adjacent side. They're also equal to the central angles in regular polygons.
Key Properties
- •Formula: β = 360° / n
- •Sum of all exterior angles: always 360°
- •Supplementary to interior angle (sum = 180°)
- •Equals central angle in regular polygons
Formulas in Radians
Interior angle: α = (n-2)π/n
Exterior angle: β = 2π/n
Sum of interior angles: (n-2)π
Sum of exterior angles: 2π
Applications
- •Architecture and construction
- •Computer graphics and game design
- •Engineering and mechanical design
- •Art and geometric patterns