How to Calculate The Degrees of An Ngon

An ngon (n-sided polygon) is a polygon with n sides. Calculating its interior and exterior angles is fundamental in geometry and architecture. This guide explains the formulas, provides an interactive calculator, and includes practical examples.

What is an Ngon?

An ngon, or n-sided polygon, is a polygon with n sides and n vertices. The simplest ngons are:

Triangle (3 sides)
Quadrilateral (4 sides)
Pentagon (5 sides)
Hexagon (6 sides)

Regular ngons have equal sides and equal angles, while irregular ngons have unequal sides and angles. The sum of the interior angles of any ngon can be calculated using a simple formula.

Calculating Interior Angles

The sum of the interior angles of an ngon is given by the formula:

Sum of interior angles = (n - 2) × 180°

For a regular ngon, each interior angle can be calculated by dividing the sum by the number of sides:

Each interior angle = (n - 2) × 180° / n

Note: This formula applies to simple polygons where no sides intersect.

Calculating Exterior Angles

The sum of the exterior angles of any ngon is always 360°, regardless of the number of sides.

Sum of exterior angles = 360°

For a regular ngon, each exterior angle can be calculated by dividing 360° by the number of sides:

Each exterior angle = 360° / n

Exterior angles are useful for determining the angle between one side and the extension of an adjacent side.

Worked Examples

Example 1: Pentagon

For a regular pentagon (n=5):

Sum of interior angles: (5 - 2) × 180° = 540°
Each interior angle: 540° / 5 = 108°
Each exterior angle: 360° / 5 = 72°

Example 2: Octagon

For a regular octagon (n=8):

Sum of interior angles: (8 - 2) × 180° = 1080°
Each interior angle: 1080° / 8 = 135°
Each exterior angle: 360° / 8 = 45°

Angle Comparison for Regular Ngons
Sides (n)	Sum of Interior Angles	Each Interior Angle	Each Exterior Angle
3 (Triangle)	180°	60°	120°
4 (Quadrilateral)	360°	90°	90°
5 (Pentagon)	540°	108°	72°
6 (Hexagon)	720°	120°	60°

FAQ

What is the difference between interior and exterior angles?

Interior angles are the angles inside the polygon at each vertex. Exterior angles are the angles formed by one side and the extension of an adjacent side.

Can I use these formulas for irregular ngons?

The sum formulas work for any simple ngon, but individual angle calculations require more information about the specific shape.

What if I have a polygon with intersecting sides?

The formulas only apply to simple polygons where sides do not intersect. Complex polygons require different methods.