—
sq units
60
units
120
°
60
°
—
units
—
units
720
°
9
—
sq units
60
units
120
°
60
°
—
units
—
units
720
°
9
The Polygon Calculator computes the area, perimeter, interior angle, exterior angle, apothem, circumradius, angle sum, and number of diagonals for any regular polygon — a polygon with all sides equal and all interior angles equal. From an equilateral triangle (\( n = 3 \)) to shapes with hundreds of sides, this calculator provides comprehensive geometric data for any regular \( n \)-gon.
Regular polygons are among the most studied objects in geometry, with a history stretching from ancient civilizations to modern mathematics. The ancient Greeks were fascinated by which regular polygons could be constructed with compass and straightedge alone. Euclid showed constructions for \( n = 3, 4, 5, 6, 15 \) and their doublings. It was not until 1796 that the teenage Carl Friedrich Gauss proved the constructibility of the regular 17-gon, a landmark result that launched modern algebra.
The area of a regular \( n \)-gon with side length \( s \) is given by \( A = \frac{ns^2}{4\tan(\pi/n)} \). This formula is derived by dividing the polygon into \( n \) isosceles triangles, each with base \( s \) and height equal to the apothem \( a = \frac{s}{2\tan(\pi/n)} \). The total area is \( A = \frac{1}{2} \times \text{perimeter} \times \text{apothem} = \frac{1}{2} \cdot ns \cdot a \).
The apothem is the distance from the center of the polygon to the midpoint of any side. It is the inradius of the polygon (radius of the inscribed circle). The circumradius is the distance from the center to any vertex (radius of the circumscribed circle), given by \( R = \frac{s}{2\sin(\pi/n)} \).
The interior angle of a regular \( n \)-gon is \( \frac{(n-2) \cdot 180°}{n} \). As \( n \) increases, this angle approaches 180°, and the polygon increasingly resembles a circle. The exterior angle is \( \frac{360°}{n} \), and the exterior angles of any convex polygon always sum to 360°. The sum of interior angles is \( (n-2) \times 180° \), a fundamental result proved by triangulating the polygon from one vertex.
The number of diagonals is \( \frac{n(n-3)}{2} \). A triangle has 0 diagonals, a quadrilateral has 2, a pentagon has 5, a hexagon has 9, and so on. This formula counts the total number of line segments connecting non-adjacent vertices.
Regular polygons have profound applications in architecture (hexagonal floor tiles, octagonal buildings), engineering (bolt patterns, gear teeth), computer graphics (mesh generation, shape approximation), nature (honeycomb cells, crystal structures), and pure mathematics (group theory, tessellation theory). The regular hexagon is especially important because it is the only regular polygon that tessellates the plane by itself (along with the equilateral triangle and square) while providing the maximum area for a given perimeter among tessellating shapes — which is why bees use hexagons.
As \( n \to \infty \), the regular polygon approaches a circle. The area formula converges to \( \pi r^2 \) and the perimeter converges to \( 2\pi r \), providing a natural bridge between polygon geometry and circle geometry. Archimedes used this convergence with 96-sided polygons to approximate \( \pi \) as between \( 3\frac{10}{71} \) and \( 3\frac{1}{7} \).
For a regular polygon with \( n \) sides of length \( s \):
Area:
$$A = \frac{n s^2}{4 \tan(\pi / n)}$$
Perimeter:
$$P = n \cdot s$$
Interior Angle:
$$\theta = \frac{(n - 2) \cdot 180°}{n}$$
Exterior Angle:
$$\phi = \frac{360°}{n}$$
Apothem (Inradius):
$$a = \frac{s}{2 \tan(\pi / n)}$$
Circumradius:
$$R = \frac{s}{2 \sin(\pi / n)}$$
Sum of Interior Angles:
$$S = (n - 2) \times 180°$$
Number of Diagonals:
$$D = \frac{n(n - 3)}{2}$$
The area gives the total surface enclosed by the polygon. The perimeter is the total boundary length. The interior angle is the angle at each vertex (all equal for a regular polygon). The apothem is the distance from the center to the nearest point on any side, useful for tiling and packing calculations. The circumradius is the distance from the center to any vertex, defining the circumscribed circle. The number of diagonals counts all internal line segments connecting non-adjacent vertices. As \( n \) increases, the polygon approaches a circle in all measurements.
Inputs
Results
A regular hexagon with side 10. The circumradius equals the side length (a unique property of regular hexagons). Area is approximately 259.81 sq units.
Inputs
Results
A regular pentagon with side 8. The interior angle is 108°, related to the golden ratio through the pentagon's diagonal-to-side ratio of φ = (1+√5)/2.
Yes, for any integer \( n \geq 3 \). The formulas apply to triangles (\( n=3 \)), squares (\( n=4 \)), pentagons (\( n=5 \)), and so on up to polygons with thousands of sides. As \( n \) increases, the polygon increasingly approximates a circle.
The apothem is the distance from the center to the midpoint of a side (inradius), while the circumradius is the distance from the center to a vertex. The apothem is always shorter than the circumradius. They are related by \( a = R \cos(\pi/n) \).
For a regular hexagon, \( R = s / (2\sin(\pi/6)) = s / (2 \times 0.5) = s \). This unique property means six equilateral triangles fit perfectly inside a hexagon, which is why hexagonal patterns are so common in nature and engineering.
As \( n \to \infty \) with fixed circumradius \( R \), the polygon area \( \frac{nR^2 \sin(2\pi/n)}{2} \) converges to \( \pi R^2 \). Archimedes used this principle with 96-sided polygons to compute one of the earliest accurate approximations of \( \pi \).
Only three regular polygons tessellate the Euclidean plane by themselves: the equilateral triangle (interior angle 60°), the square (90°), and the regular hexagon (120°). These are the only cases where the interior angle divides 360° evenly. Combinations of different regular polygons can create additional semi-regular (Archimedean) tessellations.
For irregular polygons (unequal sides and angles), the area is typically computed using the Shoelace formula given vertex coordinates: \( A = \frac{1}{2}|\sum(x_i y_{i+1} - x_{i+1} y_i)| \). This calculator is designed specifically for regular polygons where all sides and angles are equal.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
