Regular Polygon Calculator
Calculate area, perimeter, interior angles, and apothem of any regular polygon. Enter values for instant results with step-by-step formulas.
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Adjust values & calculateFormula
Where n = number of sides, s = side length. The apothem = s / (2 x tan(pi/n)), circumradius = s / (2 x sin(pi/n)), interior angle = (n-2) x 180 / n, and diagonals = n(n-3)/2.
Last reviewed: December 2025
Worked Examples
Example 1: Regular Hexagon with Side Length 10
Example 2: Regular Pentagon with Side Length 7
Background & Theory
The Regular Polygon 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 Regular Polygon 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
Sources & References
Formula
Area = (1/2) x Perimeter x Apothem = (n x s^2) / (4 x tan(pi/n))
Where n = number of sides, s = side length. The apothem = s / (2 x tan(pi/n)), circumradius = s / (2 x sin(pi/n)), interior angle = (n-2) x 180 / n, and diagonals = n(n-3)/2.
Worked Examples
Example 1: Regular Hexagon with Side Length 10
Problem: Find the area, perimeter, apothem, and interior angle of a regular hexagon with side length 10 units.
Solution: Interior angle = (6 - 2) x 180 / 6 = 120 degrees\nPerimeter = 6 x 10 = 60 units\nApothem = 10 / (2 x tan(pi/6)) = 10 / (2 x 0.5774) = 8.6603 units\nArea = (1/2) x 60 x 8.6603 = 259.8076 square units\nCircumradius = 10 / (2 x sin(pi/6)) = 10 / 1.0 = 10 units
Result: Area: 259.8076 sq units | Perimeter: 60 units | Apothem: 8.6603 units | Interior Angle: 120 degrees
Example 2: Regular Pentagon with Side Length 7
Problem: Calculate all geometric properties of a regular pentagon with side length 7 units.
Solution: Interior angle = (5 - 2) x 180 / 5 = 108 degrees\nPerimeter = 5 x 7 = 35 units\nApothem = 7 / (2 x tan(pi/5)) = 7 / (2 x 0.7265) = 4.8163 units\nArea = (1/2) x 35 x 4.8163 = 84.2854 square units\nDiagonals = 5 x (5 - 3) / 2 = 5
Result: Area: 84.2854 sq units | Perimeter: 35 units | Apothem: 4.8163 units | Diagonals: 5
Frequently Asked Questions
What is a regular polygon and how is it different from an irregular polygon?
A regular polygon is a closed two-dimensional shape where all sides have equal length and all interior angles are equal in measure. This symmetry makes regular polygons highly predictable and easy to calculate. An irregular polygon, by contrast, can have sides of different lengths and angles of different measures, making calculations much more complex. Common regular polygons include equilateral triangles, squares, regular pentagons, and regular hexagons. Regular polygons are found extensively in nature, architecture, and engineering because their symmetry distributes forces evenly and creates aesthetically pleasing designs.
How do you calculate the interior angle of a regular polygon?
The interior angle of a regular polygon is calculated using the formula (n - 2) times 180 divided by n, where n is the number of sides. This formula works because any polygon can be divided into (n - 2) triangles, and each triangle contains 180 degrees. For example, a regular hexagon with 6 sides has interior angles of (6 - 2) times 180 divided by 6, which equals 120 degrees. As the number of sides increases, the interior angle approaches but never reaches 180 degrees, which is why polygons with many sides begin to look like circles.
What is the apothem and why is it important in polygon calculations?
The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any side. It is critically important because it provides the simplest way to calculate the area of a regular polygon using the formula Area equals one-half times perimeter times apothem. The apothem is also the radius of the largest circle that can be inscribed inside the polygon, known as the incircle. For a regular polygon with side length s and n sides, the apothem equals s divided by 2 times the tangent of pi divided by n. Engineers and architects frequently use the apothem when designing hexagonal tiles, bolt patterns, and other symmetric structures.
How do you find the area of a regular polygon?
The area of a regular polygon can be calculated using the formula A equals one-half times perimeter times apothem. First compute the perimeter by multiplying the number of sides by the side length, then compute the apothem using s divided by 2 times tangent of pi over n. Alternatively, you can use A equals (n times s squared) divided by (4 times tangent of pi over n), which combines both steps into a single formula. For example, a regular hexagon with side length 10 has an area of approximately 259.81 square units. This formula works because the polygon can be divided into n identical isosceles triangles radiating from the center, each with base s and height equal to the apothem.
What is the circumradius versus the inradius of a regular polygon?
The circumradius (also called simply the radius) is the distance from the center of the polygon to any vertex, while the inradius is the distance from the center to the midpoint of any side, which is the same as the apothem. The circumradius defines the circumscribed circle that passes through all vertices, while the inradius defines the inscribed circle that is tangent to all sides. The circumradius is always larger than the inradius. For a regular polygon with side length s and n sides, the circumradius equals s divided by 2 times sine of pi over n, and the inradius equals s divided by 2 times tangent of pi over n. The ratio between these two values depends on the number of sides.
How many diagonals does a regular polygon have?
The number of diagonals in any polygon with n sides is given by the formula n times (n minus 3) divided by 2. A triangle has zero diagonals since no vertex can connect to a non-adjacent vertex. A square has 2 diagonals, a pentagon has 5, a hexagon has 9, and a decagon has 35. The formula works because each vertex can connect to n minus 3 other vertices (excluding itself and its two adjacent vertices), giving n times (n minus 3) connections, then dividing by 2 to avoid counting each diagonal twice. In regular polygons, diagonals have interesting symmetry properties and create smaller regular polygons at their intersections.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy