Polygon Exterior Angle Calculator
Solve polygon exterior angle problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
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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.
Last reviewed: December 2025
Worked Examples
Example 1: Regular Octagon Exterior Angles
Example 2: Finding Sides from Exterior Angle
Background & Theory
The Polygon Exterior Angle Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Polygon Exterior Angle Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
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\nEach exterior angle = 360 / n = 360 / 8 = 45 degrees\nEach interior angle = 180 - 45 = 135 degrees\nSum of all exterior angles = 360 degrees (always)\nSum 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\nThis is a regular pentadecagon (15-gon)\nEach interior angle = 180 - 24 = 156 degrees\nSum of interior angles = (15-2) * 180 = 2340 degrees\nNumber 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.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy