Exterior Angle of a Polygon Calculator: Understand the Formula and Examples
ByMuhammad Ali•Founder of KruskalCode
21:31
5 min read
Polygons are fundamental shapes in geometry, and understanding their angles is key to mastering the subject. While interior angles often get most of the attention, exterior angles are just as important and have a fascinating property: their sum is always the same, no matter how many sides the polygon has! This guide will walk you through what an exterior angle is, how to calculate it for regular polygons, and why their sum is always 360 degrees.
Explanation
An exterior angle of a polygon is formed when you extend one of its sides. The angle between this extended side and the adjacent side of the polygon is the exterior angle. Each vertex of a polygon has one interior angle and one exterior angle, and these two angles always add up to 180 degrees (they form a linear pair). For a regular polygon, all interior angles are equal, and consequently, all exterior angles are also equal. This makes calculating them straightforward. The total turn you make when going around the perimeter of any convex polygon is 360 degrees. Since each exterior angle represents a 'turn' at a vertex, the sum of all exterior angles must be 360 degrees. For a regular polygon with 'n' sides, this 360-degree total is divided equally among the 'n' exterior angles.
Formula
The formula for calculating a single exterior angle of a regular polygon is: Exterior Angle = 360° / n Where: 'n' = the number of sides of the regular polygon. For any convex polygon (regular or irregular), the sum of all exterior angles is always 360°.
Example
Let's say you need to find the exterior angle of a regular pentagon. A pentagon has 5 sides, so n = 5. Using the formula: Exterior Angle = 360° / 5 Exterior Angle = 72° So, each exterior angle of a regular pentagon is 72 degrees. If you were to add all five exterior angles (72° * 5), you would get 360°, confirming the sum property.
How to use the related calculator
Using our Exterior Angle of a Polygon Calculator is simple. Just locate the input field labeled 'Number of Sides (n)'. Enter the total number of sides your regular polygon has (e.g., 3 for a triangle, 4 for a square, 8 for an octagon). Make sure you enter a whole number greater than 2. Once you've entered the value, the calculator will instantly display the measure of each individual exterior angle and remind you that the sum of all exterior angles is always 360 degrees.
Try the related calculator
Open toolFAQ
What's the difference between an interior and an exterior angle?
An interior angle is inside the polygon, formed by two adjacent sides. An exterior angle is outside the polygon, formed by one side and the extension of an adjacent side. They are supplementary, meaning they add up to 180 degrees.
Does the sum of exterior angles being 360 degrees apply to all polygons?
Yes, this rule applies to all convex polygons, whether they are regular (all sides and angles equal) or irregular (sides and angles can be different). It's a fundamental property of polygons.
Can I calculate the exterior angle if I only know the interior angle?
Absolutely! Since an interior angle and its adjacent exterior angle form a linear pair and sum to 180 degrees, you can find the exterior angle by subtracting the interior angle from 180 degrees. For example, if an interior angle is 108°, the exterior angle is 180° - 108° = 72°.
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About the author
Muhammad Ali. Muhammad Ali is a full-stack developer and founder of KruskalCode. He builds SaaS platforms and automation systems with React and Laravel, and helps teams ship fast, scalable tools.
Need a custom calculator, dashboard, or automation workflow? Reach out to KruskalCode.
