Polygon Angle Calculator
Calculate interior and exterior angles for any regular polygon. Enter the number of sides to find each angle, sum of angles, number of diagonals, and triangles formed.
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Minimum 3 sides (triangle)
Result
Enter values above and click Calculate to see your result.
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Formula
How it works: A regular polygon with n sides has an interior angle sum of (n - 2) x 180 degrees. Each interior angle equals this sum divided by n. Each exterior angle is 360/n degrees, and the sum of all exterior angles is always 360 degrees.
Worked Example
What Are Polygon Angles?
A polygon is a closed two-dimensional shape made up of straight line segments. When all sides are equal in length and all interior angles are equal in measure, it is called a regular polygon.
Understanding polygon angles is fundamental in geometry and various practical applications. Each regular polygon possesses specific interior and exterior angles that can be systematically calculated.
The interior angle is the angle inside the polygon formed by two adjacent sides, while the exterior angle is formed by one side and an extension of an adjacent side. A key property is that an interior angle and its corresponding exterior angle always sum to 180 degrees.
The sum of all interior angles for a polygon with 'n' sides is given by the formula (n - 2) * 180 degrees, and each individual interior angle is this sum divided by 'n'. Interestingly, the sum of all exterior angles for any convex polygon, regular or irregular, is always 360 degrees.
These calculations are crucial not only for academic study but also in fields like architecture, engineering, and computer graphics for designing structures and rendering shapes.
- Regular polygons have equal side lengths and equal angle measures.
- Interior and exterior angles at any vertex sum to 180 degrees.
- The sum of all exterior angles for any convex polygon is always 360 degrees.
- Polygons can be divided into triangles to determine the sum of their interior angles.
Grasping these angle relationships helps in solving complex geometric problems and designing precise structures. Use the Polygon Angle Calculator to effortlessly explore these properties for any polygon.
You can also calculate changes using our Circle Geometry Theorem Solver, Semicircle Calculator or Midpoint and Distance Calculator.
Frequently Asked Questions
What is the interior angle of a decagon?
A regular decagon (10 sides) has interior angles of 144 degrees each. The sum of all interior angles is 1,440 degrees.
What is the interior angle of a hexagon?
A regular hexagon (6 sides) has interior angles of 120 degrees each. The sum is (6-2) x 180 = 720 degrees.
Why do exterior angles always sum to 360?
If you walk around any convex polygon, you make one complete turn (360 degrees). Each exterior angle represents part of that turn, so they must sum to exactly 360 degrees regardless of the number of sides.
How can I put this Polygon Angle Calculator on my blog or website?
Yes, the Polygon Angle Calculator is fully embeddable. Tap "Embed" above to configure appearance and copy the code. It is free to use, works on any platform (HTML, WordPress, CMS), and adjusts to any screen size automatically. Visit calculory.com/services/embed-calculators for the complete guide.
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