Polygon Interior Angle Calculator
Calculate polygon interior angle instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
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Where n is the number of sides. The sum of interior angles equals (n-2) times 180 degrees because the polygon can be divided into (n-2) triangles. For regular polygons, divide by n to get each individual angle.
Last reviewed: December 2025
Worked Examples
Example 1: Regular Pentagon Interior Angles
Example 2: Finding Sides from Interior Angle
Background & Theory
The Polygon Interior 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 Interior 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
Sum = (n - 2) * 180 | Each = (n - 2) * 180 / n
Where n is the number of sides. The sum of interior angles equals (n-2) times 180 degrees because the polygon can be divided into (n-2) triangles. For regular polygons, divide by n to get each individual angle.
Worked Examples
Example 1: Regular Pentagon Interior Angles
Problem: Find each interior angle and total angle sum of a regular pentagon.
Solution: Number of sides n = 5\nSum of interior angles = (5-2) * 180 = 3 * 180 = 540 degrees\nEach interior angle = 540 / 5 = 108 degrees\nEach exterior angle = 180 - 108 = 72 degrees\nNumber of diagonals = 5(5-3)/2 = 5\nTriangles formed = 5 - 2 = 3
Result: Each interior angle = 108 degrees | Sum = 540 degrees
Example 2: Finding Sides from Interior Angle
Problem: A regular polygon has each interior angle measuring 150 degrees. How many sides?
Solution: Exterior angle = 180 - 150 = 30 degrees\nNumber of sides = 360 / 30 = 12\nThis is a regular dodecagon\nSum of interior angles = (12-2) * 180 = 1800 degrees\nVerification: 1800 / 12 = 150 degrees per angle
Result: 12 sides (dodecagon) | Sum of angles = 1800 degrees
Frequently Asked Questions
What is the interior angle sum formula for polygons?
The interior angle sum formula states that the sum of all interior angles of a polygon with n sides equals (n - 2) times 180 degrees. This formula works because any polygon can be divided into (n - 2) non-overlapping triangles by drawing diagonals from a single vertex, and each triangle contributes 180 degrees to the total angle sum. For a triangle (n=3), the sum is 180 degrees. For a quadrilateral (n=4), it is 360 degrees. For a pentagon (n=5), it is 540 degrees. This formula applies to both regular and irregular polygons, convex and concave, as long as the polygon is simple (non-self-intersecting).
How do you find each interior angle of a regular polygon?
For a regular polygon where all sides and angles are equal, each interior angle equals the sum of interior angles divided by the number of sides: ((n - 2) * 180) / n degrees. This simplifies to 180 - 360/n, which shows that each interior angle equals 180 degrees minus the exterior angle. A regular triangle has angles of 60 degrees, a square has 90 degrees, a regular pentagon has 108 degrees, and a regular hexagon has 120 degrees. As the number of sides increases, each interior angle approaches but never reaches 180 degrees, with the polygon increasingly resembling a circle.
What is the difference between interior and exterior angles?
Interior angles are formed inside the polygon between two adjacent sides. Exterior angles are formed between one side and the extension of the adjacent side, lying outside the polygon. At each vertex, the interior and exterior angles are supplementary, summing to 180 degrees. While the sum of interior angles increases with the number of sides ((n-2)*180), the sum of exterior angles is always 360 degrees regardless of the polygon type. This makes exterior angles simpler for certain calculations, while interior angles are more intuitive for understanding the shape and for construction purposes.
How do you find the number of sides from an interior angle?
If you know the interior angle of a regular polygon, you can find the number of sides using the formula n = 360 / (180 - interior angle). First compute the exterior angle as 180 minus the interior angle, then divide 360 by this exterior angle. For example, if the interior angle is 144 degrees, the exterior angle is 36 degrees, and n = 360/36 = 10 (decagon). The result must be a whole number of 3 or greater for a valid regular polygon. If the calculation yields a non-integer, then no regular polygon has that exact interior angle measure.
What are diagonals of a polygon and how many are there?
A diagonal is a line segment connecting two non-adjacent vertices of a polygon. The number of diagonals in an n-sided polygon is n(n-3)/2. This formula comes from choosing 2 vertices from n (which gives n(n-1)/2 line segments) and subtracting the n sides. A triangle has 0 diagonals, a quadrilateral has 2, a pentagon has 5, a hexagon has 9, and a decagon has 35. Diagonals are used to triangulate polygons, which is essential for computing areas, for rendering polygons in computer graphics, and for many computational geometry algorithms.
Why can a polygon be divided into triangles using diagonals?
Any simple polygon with n vertices can be triangulated into exactly (n-2) non-overlapping triangles by drawing (n-3) non-crossing diagonals from a single vertex. This is possible because a diagonal from one vertex to a non-adjacent vertex splits the polygon into two smaller polygons, and the process can be repeated recursively. The triangulation is not unique (there are typically many valid triangulations), but the number of resulting triangles is always n-2. This property is the foundation of the interior angle sum formula and is heavily used in computational geometry, finite element methods, and 3D rendering pipelines.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy