Calculating Angles in Regular Polygons: Formulas and Examples

Finding the size of each angle in a regular polygon is a fundamental concept in geometry. This article provides step-by-step instructions on how to calculate the interior and exterior angles of a regular polygon using mathematical formulas. We will also explore the relationship between the number of sides of a polygon and the measures of its angles. By the end of this guide, you will be able to easily determine the angles of any regular polygon.

The Formulas

There are two primary formulas used to calculate the angles in a regular polygon:

Interior Angle: Use the formula: Interior Angle (n - 2) × 180° / n, where n is the number of sides in the polygon.

Exterior Angle: Use the formula: Exterior Angle 360° / n.

Additionally, the interior angle can be calculated using the exterior angle or the central angle:

Exterior Angle (D) Interior Angle Central Angle (C)

Exterior Angle (E) 360° / n

Interior Angle (I) 180° - Exterior Angle

Central Angle (C) Exterior Angle

How to Calculate the Angles

Let's break down the steps:

Identify the Number of Sides (n): Count the number of sides in your polygon.

Substitute the Value into the Formulas: Plug the value of n into the appropriate formulas to find the measures of the angles.

Example Calculations

Let's work through a few examples to illustrate the process:

Triangle (n 3):

Using the formula, Interior Angle (3 - 2) × 180° / 3, we get:

Interior Angle 60°

Alternatively, the exterior angle is 360° / 3 120°, and the interior angle is 180° - 120° 60°.

Square (n 4):

Using the formula, Interior Angle (4 - 2) × 180° / 4, we get:

Interior Angle 90°

The exterior angle is 360° / 4 90°, which means the interior angle is 180° - 90° 90°.

Pentagon (n 5):

Using the formula, Interior Angle (5 - 2) × 180° / 5, we get:

Interior Angle 108°

The exterior angle is 360° / 5 72°, and the interior angle is 180° - 72° 108°.

Nonagon (n 9):

Using the exterior angle formula, Exterior Angle 360° / 9, we get:

Exterior Angle 40°

The interior angle is 180° - 40° 140°, and the central angle (which is the same as the exterior angle) is 40°.

Dodecagon (n 12):

Using the formula, Exterior Angle 360° / 12, we get:

Exterior Angle 30°

The interior angle is 180° - 30° 150°, and the central angle (which is the same as the exterior angle) is 30°.

Conclusion

Understanding how to calculate the angles in a regular polygon is crucial for a wide range of applications, from basic geometry to more advanced fields of mathematics. Whether you are a student, teacher, or professional in a related field, these formulas and examples will help you master the process. By following the steps outlined in this guide, you can now easily find the measures of any regular polygon's angles.

Additional Resources

For those who want to learn more about polygons and their properties, here are some additional resources:

Math is Fun - Polygons

Wikipedia - Polygon

Math Planet - Calculation of Interior and Exterior Angles

These resources provide comprehensive information on polygons, including definitions, properties, and more detailed calculations.