Polygon Exterior Angle Calculator
Solve polygon exterior angle problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Polygon Exterior Angle Calculator
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Formula: Exterior Angle = 360 / n
Worked example — Each exterior angle = 45 degrees | Interior angle = 135 degrees
Formula
Exterior Angle = 360 / n
Where n is the number of sides of the regular polygon. The sum of all exterior angles of any convex polygon is always 360 degrees. Each exterior angle and its corresponding interior angle sum to 180 degrees.
Worked Examples
Example 1: Regular Octagon Exterior Angles
Problem:Find the measure of each exterior angle of a regular octagon.
Solution:Number of sides n = 8 Each exterior angle = 360 / n = 360 / 8 = 45 degrees Each interior angle = 180 - 45 = 135 degrees Sum of all exterior angles = 360 degrees (always) Sum of all interior angles = (8-2) * 180 = 1080 degrees
Result:Each exterior angle = 45 degrees | Interior angle = 135 degrees
Example 2: Finding Sides from Exterior Angle
Problem:A regular polygon has each exterior angle measuring 24 degrees. How many sides does it have?
Solution:Number of sides = 360 / exterior angle = 360 / 24 = 15 This is a regular pentadecagon (15-gon) Each interior angle = 180 - 24 = 156 degrees Sum of interior angles = (15-2) * 180 = 2340 degrees Number of diagonals = 15(15-3)/2 = 90
Result:15 sides (pentadecagon) | Interior angle = 156 degrees
Frequently Asked Questions
What is an exterior angle of a polygon?
An exterior angle of a polygon is formed between one side of the polygon and the extension of an adjacent side. At each vertex, the exterior angle and the interior angle are supplementary, meaning they add up to 180 degrees. If you walk along the perimeter of any convex polygon, turning at each vertex by the exterior angle, you complete exactly one full rotation of 360 degrees by the time you return to the starting point. This fundamental property holds regardless of the number of sides. For a regular polygon (all sides and angles equal), each exterior angle has the same measure, making the calculation straightforward.
Why do exterior angles of any convex polygon always sum to 360 degrees?
The exterior angle sum theorem states that the sum of exterior angles of any convex polygon equals 360 degrees, regardless of the number of sides. This can be understood intuitively by imagining walking along the polygon perimeter. At each vertex you turn by the exterior angle, and after traversing all vertices you face the same direction as when you started, having turned through one complete revolution. Algebraically, since each interior-exterior pair sums to 180 degrees, the total is n times 180 minus the sum of interior angles, which equals n times 180 minus (n-2) times 180, giving exactly 360 degrees.
How do you calculate each exterior angle of a regular polygon?
For a regular polygon with n sides, each exterior angle equals 360 divided by n degrees. This follows directly from the fact that all exterior angles are equal in a regular polygon and their sum is 360 degrees. For example, a regular triangle has exterior angles of 360/3 = 120 degrees each. A regular hexagon has 360/6 = 60 degrees each. A regular decagon has 360/10 = 36 degrees each. This formula also works in reverse: if you know the exterior angle, divide 360 by it to find the number of sides. An exterior angle of 45 degrees means the polygon has 360/45 = 8 sides (regular octagon).
What is the relationship between interior and exterior angles?
At each vertex of a polygon, the interior angle and the exterior angle are supplementary angles, meaning they sum to exactly 180 degrees. This relationship provides a quick conversion between the two: exterior angle = 180 minus interior angle, and vice versa. For a regular hexagon with interior angles of 120 degrees, each exterior angle is 180 - 120 = 60 degrees. This supplementary relationship exists because the interior angle and exterior angle together form a straight line (the extension of one side). The relationship is useful for solving geometry problems where one type of angle is given and the other is needed.
How are exterior angles used in navigation and surveying?
In land surveying, exterior angles are measured when traversing property boundaries. A surveyor walks along each boundary line and measures the turning angle at each corner, which is the exterior angle. The sum should be 360 degrees for a closed traverse, providing a built-in error check. In navigation, exterior angles correspond to course changes at waypoints. Pilots and sailors use turning angles (exterior angles) to plot routes. Robot path planning also uses exterior angles to determine how much to turn at each corner of a polygonal path. These practical applications rely on the constant 360-degree sum property.
What happens with exterior angles of concave polygons?
For concave (non-convex) polygons, some exterior angles become negative because the interior angle exceeds 180 degrees at reflex vertices. At a reflex vertex, the polygon bends inward, and the exterior angle is measured as a negative turn. Even with these negative angles, the signed sum of all exterior angles still equals 360 degrees for a simple (non-self-intersecting) polygon. This generalized version of the exterior angle sum theorem uses the concept of signed angles and is related to the winding number in topology. Understanding signed exterior angles is important for computational geometry algorithms that process arbitrary polygonal shapes.
How do exterior angles change as the number of sides increases?
As a regular polygon gains more sides, each exterior angle decreases because the fixed total of 360 degrees is divided among more angles. A triangle has 120-degree exterior angles, a square has 90 degrees, a hexagon has 60 degrees, and a 360-gon has just 1-degree exterior angles. In the limit, as the number of sides approaches infinity, the exterior angle approaches zero and the polygon approaches a circle. This connection between polygons and circles is fundamental in geometry and is used in approximation algorithms. Archimedes used this principle to estimate pi by inscribing and circumscribing polygons with increasing numbers of sides.
Can you determine the number of sides from the exterior angle?
Yes, if a regular polygon has a known exterior angle measure, the number of sides equals 360 divided by the exterior angle. However, the result must be a whole number of 3 or greater for a valid polygon. An exterior angle of 72 degrees gives 360/72 = 5 sides (pentagon). An exterior angle of 40 degrees gives 360/40 = 9 sides (nonagon). If dividing 360 by the angle does not produce an integer, then no regular polygon has that exact exterior angle measure. This method is frequently used in standardized math tests and geometry competitions to identify polygons from angle measurements.
What is the connection between exterior angles and central angles?
For a regular polygon inscribed in a circle, the central angle (the angle at the center subtended by one side) equals the exterior angle. Both are 360 divided by n for an n-sided regular polygon. This is not a coincidence but follows from the geometry of inscribed polygons. The central angle represents the arc each side subtends, while the exterior angle represents the turning at each vertex, and both divide the full 360-degree rotation equally. This connection is used in circle geometry, where regular polygons are constructed by placing vertices at equally spaced points around a circle using central angles.
How are exterior angles used in tessellations and tiling?
Exterior angles help determine which regular polygons can tile a flat surface (tessellate). For a regular tessellation, the interior angles meeting at each vertex must sum to exactly 360 degrees. Since the interior angle equals 180 minus the exterior angle, this condition constrains which polygons work. Only equilateral triangles (interior 60 degrees), squares (90 degrees), and regular hexagons (120 degrees) can form regular tessellations by themselves, because 360 is divisible by 60, 90, and 120. Semi-regular tessellations use combinations of different regular polygons, and the exterior angle relationships determine which combinations fit together without gaps.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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