How Calculate Degrees Angles in N-Gon

An n-gon is a polygon with n sides. Calculating the interior and exterior angles of an n-gon is essential in geometry, architecture, and design. This guide explains the formulas, provides an interactive calculator, and includes practical examples.

What is an N-gon?

An n-gon is a polygon with n sides and n vertices. The most common polygons are:

Triangle (3-gon)
Quadrilateral (4-gon)
Pentagon (5-gon)
Hexagon (6-gon)
Heptagon (7-gon)
Octagon (8-gon)

Regular polygons have all sides and angles equal, while irregular polygons have varying side lengths and angles. The sum of the interior angles of any n-gon can be calculated using a simple formula.

Calculating Interior Angles

The sum of the interior angles of an n-sided polygon is given by:

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

For a regular n-gon, each interior angle is:

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

Example Calculation

For a regular pentagon (5-gon):

Pentagon Interior Angle Calculation

Sum of interior angles = (5 - 2) × 180° = 540°

Each interior angle = 540° / 5 = 108°

Calculating Exterior Angles

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

Sum of exterior angles = 360°

For a regular n-gon, each exterior angle is:

Each exterior angle = 360° / n

Example Calculation

For a regular hexagon (6-gon):

Hexagon Exterior Angle Calculation

Sum of exterior angles = 360°

Each exterior angle = 360° / 6 = 60°

Worked Examples

Example 1: Octagon

Calculate the interior and exterior angles of a regular octagon (8-gon).

Octagon Angle Calculation

Sum of interior angles = (8 - 2) × 180° = 1080°

Each interior angle = 1080° / 8 = 135°

Each exterior angle = 360° / 8 = 45°

Example 2: Decagon

Calculate the interior and exterior angles of a regular decagon (10-gon).

Decagon Angle Calculation

Sum of interior angles = (10 - 2) × 180° = 1440°

Each interior angle = 1440° / 10 = 144°

Each exterior angle = 360° / 10 = 36°

Comparison Table

Polygon	Sum of Interior Angles	Each Interior Angle	Each Exterior Angle
Triangle (3-gon)	180°	60°	120°
Square (4-gon)	360°	90°	90°
Pentagon (5-gon)	540°	108°	72°
Hexagon (6-gon)	720°	120°	60°

Frequently Asked Questions

What is the difference between interior and exterior angles?

Interior angles are the angles inside the polygon at each vertex. Exterior angles are formed by extending one side of the polygon and measuring the angle between this extension and the adjacent side.

Can I calculate angles for irregular polygons?

Yes, but you'll need to know the individual side lengths and use trigonometric functions to calculate each angle separately. The formulas provided work for regular polygons only.

What's the smallest number of sides an n-gon can have?

An n-gon must have at least 3 sides (a triangle) to form a closed shape. Polygons with fewer than 3 sides are not considered polygons.

How do I verify my angle calculations?

For regular polygons, you can verify by ensuring the sum of all interior angles equals (n-2) × 180° and each exterior angle equals 360° divided by n. For irregular polygons, use a protractor to measure angles directly.