How Do You Calculate Degrees in A Polygon

Understanding how to calculate degrees in a polygon is essential for geometry, architecture, and design. Whether you're working with triangles, quadrilaterals, or complex polygons, knowing how to determine interior and exterior angles can help you solve problems in various fields.

What Are Polygon Degrees?

Polygon degrees refer to the angles formed by the sides of a polygon. There are two main types of degrees to consider:

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

Understanding these degrees is crucial for analyzing shapes, calculating areas, and solving geometric problems.

Calculating Interior Degrees

The sum of the interior degrees of a polygon can be calculated using the formula:

Interior Degrees Formula

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

Where n is the number of sides in the polygon.

To find the measure of each interior angle, divide the sum by the number of sides:

Individual Interior Angle

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

For example, a pentagon (5 sides) has interior angles that sum to (5 - 2) × 180° = 540°. Each interior angle is 540° / 5 = 108°.

Calculating Exterior Degrees

Exterior degrees are calculated by extending one side of the polygon and measuring the angle formed with an adjacent side. The sum of the exterior degrees of any polygon is always 360°.

Exterior Degrees Formula

Sum of exterior degrees = 360°

To find the measure of each exterior angle, divide the sum by the number of sides:

Individual Exterior Angle

Each exterior angle = 360° / n

For example, a hexagon (6 sides) has exterior angles that sum to 360°. Each exterior angle is 360° / 6 = 60°.

Worked Examples

Example 1: Triangle (3 sides)

Sum of interior degrees: (3 - 2) × 180° = 180°

Each interior angle: 180° / 3 = 60°

Sum of exterior degrees: 360°

Each exterior angle: 360° / 3 = 120°

Example 2: Octagon (8 sides)

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

Each interior angle: 1080° / 8 = 135°

Sum of exterior degrees: 360°

Each exterior angle: 360° / 8 = 45°

Frequently Asked Questions

What is the difference between interior and exterior degrees?

Interior degrees are the angles inside the polygon at each vertex, while exterior degrees are the angles formed by extending one side of the polygon and measuring the angle with an adjacent side.

Can I calculate the degrees of any polygon?

Yes, the formulas work for any polygon with three or more sides. Simply plug in the number of sides to calculate the degrees.

What is the sum of exterior degrees for any polygon?

The sum of exterior degrees is always 360°, regardless of the number of sides in the polygon.