Polygon Calculator
Calculate area, perimeter, apothem, circumradius, interior angles, and diagonals of any regular polygon by entering the number of sides and side length.
Polygon Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: Area = (1/2) x Perimeter x Apothem
Worked example — Area: 166.28 cm2 | Perimeter: 48 cm | Apothem: 6.93 cm
Formula
Area = (1/2) x Perimeter x Apothem
The area of a regular polygon equals half the product of its perimeter and apothem. The apothem is s / (2 tan(pi/n)) and the circumradius is s / (2 sin(pi/n)), where n is the number of sides and s is the side length.
Worked Examples
Example 1: Regular Hexagon Calculations
Problem:Find the area, perimeter, apothem, and circumradius of a regular hexagon with side length 8 cm.
Solution:Perimeter = 6 x 8 = 48 cm Apothem = 8 / (2 x tan(pi/6)) = 8 / (2 x 0.5774) = 6.9282 cm Area = 0.5 x 48 x 6.9282 = 166.28 cm2 Circumradius = 8 / (2 x sin(pi/6)) = 8 / 1.0 = 8.0 cm Interior angle = (6-2) x 180 / 6 = 120 degrees
Result:Area: 166.28 cm2 | Perimeter: 48 cm | Apothem: 6.93 cm
Example 2: Regular Octagon Properties
Problem:Calculate the properties of a regular octagon with side length 5 cm.
Solution:Perimeter = 8 x 5 = 40 cm Apothem = 5 / (2 x tan(pi/8)) = 5 / 0.8284 = 6.0355 cm Area = 0.5 x 40 x 6.0355 = 120.71 cm2 Circumradius = 5 / (2 x sin(pi/8)) = 5 / 0.7654 = 6.5328 cm Diagonals = 8(8-3)/2 = 20
Result:Area: 120.71 cm2 | Perimeter: 40 cm | Diagonals: 20
Frequently Asked Questions
What is a regular polygon and how is its area calculated?
A regular polygon is a closed two-dimensional shape where all sides are equal in length and all interior angles are equal in measure. Examples include equilateral triangles, squares, regular pentagons, and regular hexagons. The area of a regular polygon is calculated using the formula Area equals one-half times the perimeter times the apothem, where the apothem is the perpendicular distance from the center of the polygon to the midpoint of any side. Alternatively, the area can be computed as (n times s squared) divided by (4 times the tangent of pi over n), where n is the number of sides and s is the side length. This formula works for any regular polygon regardless of the number of sides.
What is the apothem and circumradius of a polygon?
The apothem and circumradius are two important measurements associated with regular polygons. The apothem is the perpendicular distance from the center of the polygon to the midpoint of any side. It is calculated as s divided by two times the tangent of pi over n, where s is the side length and n is the number of sides. The circumradius is the distance from the center of the polygon to any vertex, and it is calculated as s divided by two times the sine of pi over n. The circumradius always equals or exceeds the apothem. As the number of sides increases, the ratio between circumradius and apothem approaches one, which corresponds to the polygon approaching a circle shape.
How do you calculate the interior angles of a polygon?
The sum of interior angles of any polygon with n sides is given by the formula (n minus 2) times 180 degrees. For a regular polygon where all angles are equal, each interior angle equals this sum divided by n. So for a triangle it is 60 degrees, for a square 90 degrees, for a pentagon 108 degrees, for a hexagon 120 degrees, for an octagon 135 degrees, and for a decagon 144 degrees. The exterior angle of a regular polygon is simply 360 divided by n. Notice that interior and exterior angles always sum to 180 degrees. As the number of sides increases toward infinity, the interior angle approaches but never reaches 180 degrees, and the polygon increasingly resembles a circle.
How many diagonals does a polygon have?
The number of diagonals in a polygon with n sides is calculated using the formula n times (n minus 3) divided by 2. A diagonal is a line segment connecting two non-adjacent vertices. A triangle has zero diagonals because every vertex is adjacent to every other vertex. A quadrilateral has 2 diagonals, a pentagon has 5, a hexagon has 9, an octagon has 20, and a decagon has 35 diagonals. The formula derives from the fact that each vertex connects to n minus 3 other non-adjacent vertices through diagonals, giving n times (n minus 3) connections total, but each diagonal is counted twice so we divide by 2. Understanding diagonals is important in computational geometry and triangulation algorithms.
How does a polygon approximate a circle as sides increase?
As the number of sides of a regular polygon increases, it progressively approximates a circle. This is a fundamental concept in both geometry and calculus. With 10 sides, a regular polygon already looks quite circular. With 100 sides, it is nearly indistinguishable from a circle to the naked eye. Mathematically, as n approaches infinity, the area of a regular polygon with circumradius r approaches pi times r squared, which is the area formula for a circle. Similarly the perimeter approaches 2 times pi times r, the circumference of a circle. The ancient mathematician Archimedes used this principle to approximate pi by inscribing and circumscribing polygons around a circle and calculating their perimeters, achieving remarkable accuracy with 96-sided polygons.
What is the difference between a convex and a concave polygon?
In a convex polygon every interior angle is less than 180 degrees and every diagonal lies wholly inside the shape. A concave polygon has at least one reflex angle greater than 180 degrees, which creates a notch, and at least one diagonal that passes outside the boundary. The formulas on this page assume a convex regular polygon, so apothem, circumradius and the half-perimeter-times-apothem area all require it. The interior angle sum of (n - 2) times 180 degrees, however, holds for any simple polygon, convex or concave.
How do you find the area of an irregular polygon from its vertex coordinates?
Use the shoelace formula: list the vertices in order around the boundary, sum the cross products x1 y2 minus x2 y1 over consecutive pairs including the wrap from the last vertex back to the first, then take half the absolute value. The vertices must be traversed consistently, all clockwise or all counter-clockwise; a mixed order produces a wrong figure. The sign before you take the absolute value tells you the orientation, which is why the same expression is used in computational geometry to test whether points are listed clockwise.
Which regular polygons can tile a plane by themselves?
Only three: the equilateral triangle, the square, and the regular hexagon. The condition is that copies meeting at a vertex must fill exactly 360 degrees, so the interior angle has to divide 360 without remainder. That works for 60, 90, and 120 degrees, giving six triangles, four squares, or three hexagons at each vertex. A regular pentagon fails at 108 degrees and a regular octagon at 135, though octagons combine with squares to give the familiar truncated square tiling used on floors.
Which regular polygons can be drawn with only a compass and straightedge?
Gauss and Wantzel settled this: a regular n-gon is constructible exactly when n is a power of 2 multiplied by distinct Fermat primes, the known ones being 3, 5, 17, 257 and 65537. So the triangle, square, pentagon, hexagon, octagon, decagon, dodecagon, 15-gon and 17-gon are all constructible, while the regular heptagon, nonagon, and hendecagon are not. Their side lengths still have perfectly good values, computable from s = 2R sin(pi/n); constructibility limits the drawing method, not the arithmetic.
How do you find the side length when you know the area or the circumradius?
From the circumradius R, the side is s = 2R sin(pi/n), since each side subtends a central angle of 2pi/n. From the apothem a, it is s = 2a tan(pi/n). To go from a target area A, invert the area formula A = n s squared divided by 4 tan(pi/n) to get s = sqrt(4A tan(pi/n) divided by n). That last form is the practical one when a floor plan, tank base, or plate is specified by required area and you need to know how long to cut each edge.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮Shoelace Formula Polygon Area Calculator
Calculate shoelace formula polygon area with inputs, formulas, and instant results.
🧮Irregular Polygon Area Calculator
Calculate irregular polygon area with inputs, formulas, and instant results.
🧮Polygon Exterior Angle Calculator
Calculate polygon exterior angle with inputs, formulas, and instant results.
🧮Polygon Interior Angle Calculator
Calculate polygon interior angle with inputs, formulas, and instant results.
🧮Regular Polygon Calculator
Calculate area, perimeter, interior angles, and apothem of any regular polygon.
🧮Annulus Area Calculator
Calculate annulus area with inputs, formulas, and instant results.
🧮Area Calculator
Calculate area with inputs, formulas, and instant results.
🧮Area of a Rectangle Calculator
Calculate the area, perimeter, and diagonal of a rectangle. Find missing sides from known area. Convert between metric and imperial area units.