Regular Polygon Calculator - Area, Perimeter & Apothem

A regular polygon calculator solves the geometry of any n-sided polygon whose sides are equal in length and whose interior angles are all the same. Enter the number of sides and the length of one side, and the tool returns the perimeter, area, apothem, circumradius, interior angle, and exterior angle in real time. It is built for students, designers laying out hexagonal tiles or decagonal skylights, and engineers working with regular cross-sections.

• • Homework and exam preparation: Check answers for problems about area, perimeter, apothem, and angles of any regular polygon.
• • Architecture and tiling layouts: Compute the side length, area, and apothem for regular hexagonal tiles, octagonal windows, or decagonal skylights.
• • Mechanical and design drafting: Compute the circumradius of a bolt-circle pattern or the inradius of a regular nut profile.
• • Programming and graphics work: Generate vertex coordinates of a regular n-gon for games or graphics routines without re-deriving the trigonometry.

The math that describes a regular polygon only needs three pieces of information: the number of sides n, the side length a, and the constant pi. With those, every other measurement follows. The regular polygon calculator on this page runs those derivations for you, and it labels the polygon with its common name so you can sanity-check the result.

For projects that already know the apothem or perimeter and want to start there instead of from the side length, Polygon Area Calculator handles the same shape using a different entry value.

The regular polygon calculator combines six closed-form formulas that all depend on the number of sides n and the side length a. It labels the shape with its common polygon name and computes the interior and exterior angles in degrees, with units kept consistent across the result panel.

• n: Number of equal sides; an integer >= 3. Inputs below 3 are clamped to 3.
• a: Side length. Any non-negative real number; the same units propagate to every output.
• alpha: Interior angle at each vertex in degrees. Approaches 180 as n grows.
• beta: Exterior angle at each vertex in degrees. Approaches 0 as n grows.
• r (apothem): Inradius, the distance from the center to the middle of a side.
• R (circumradius): Circumradius, the distance from the center to a vertex.

The apothem and circumradius share the same denominator structure; the only difference is sin or tan. The apothem uses tan(pi / n) for the half-angle to the side midpoint, and the circumradius uses sin(pi / n) for the half-angle to a vertex.

According to Omni Calculator: Polygon Calculator, the area of a regular polygon with n sides of length a is A = n * a^2 / (4 * tan(pi / n)), the apothem is a / (2 * tan(pi / n)), the circumradius is a / (2 * sin(pi / n)), the interior angle is (n - 2) * 180 / n, and the exterior angle is 360 / n.

The hexagon is the most-used case of the regular polygon formulas, and Hexagon Calculator focuses on that specific shape with dedicated apothem, perimeter, and diagonal outputs.

Four ideas make the regular polygon formulas work: every regular polygon is built from congruent isosceles triangles, the side length controls everything, the apothem and circumradius are different center-to-edge distances, and the angles sum to fixed amounts.

A handy way to see why these formulas work is to draw lines from the center to each vertex: you get n congruent isosceles triangles, each with two sides of length R and a base of length a. The area of one triangle is (1/2) * R^2 * sin(2 * pi / n), and multiplying by n gives the area formula expressed through R instead of a.

The smallest valid regular polygon is the equilateral triangle, and Triangle Calculator solves the same set of measurements plus extras like the inradius and circumradius formulas for any triangle, not just the equilateral one.

The interface has two inputs and seven live results, so the workflow is short. The polygon name updates as the number of sides changes.

1. 1 Enter the number of sides: Type the integer n in the first field. Use 3 for an equilateral triangle, 4 for a square, 6 for a regular hexagon, 8 for an octagon, and so on. The default is 6.
2. 2 Enter the side length: Type the length of one side in the second field. Use whatever unit you are working in. The default is 4.
3. 3 How to use the polygon name: The first result confirms the shape: Regular hexagon for n = 6 or Regular decagon for n = 10.
4. 4 How to use the perimeter and area: Perimeter is the boundary length and area is the surface covered. Use these to size materials or check answers.
5. 5 How to use the apothem and circumradius: Apothem r is the center-to-side distance, useful for incircles. Circumradius R is the center-to-vertex distance, useful for bolt circles.
6. 6 How to use the interior and exterior angles: Interior angle alpha and exterior angle beta are displayed in degrees. They sum to 180 for any n. Use alpha for vertex bevels and beta for layout.

Try a worked example: enter n = 6 and side length a = 4. The result panel shows a Regular hexagon with perimeter 24, area 41.5692 sq units, apothem 3.4641, circumradius 4, interior angle 120 deg, and exterior angle 60 deg.

As n grows past a few hundred sides the regular polygon is visually indistinguishable from a circle, and Circle Calculator is the natural next step when the user wants to drop the side constraint and reason about radius and circumference directly.

Hand-calculating regular polygon values is fine for triangles and squares but becomes error-prone as n grows. The calculator removes that error class and bundles the results in one place.

• • Six outputs from two inputs: Perimeter, area, apothem, circumradius, interior angle, and exterior angle are all derived from n and a. No need to look up six different formulas.
• • Works for any n: From triangle (n = 3) to 1000-gon and beyond, the same code path computes the result. The interface labels the shape with its common name when n <= 12.
• • Real-time updates: Each result recomputes as you type, with a short animation that signals the change.
• • Two equivalent area formulas: The same area is computed in two ways: A = n * a^2 / (4 * tan(pi / n)) and A = (1/2) * apothem * perimeter. They agree to floating-point precision, which makes the calculator a useful cross-check.
• • Sanity checks built in: Edge cases (n below 3, non-integer n, side length 0) are clamped or returned as zero so the result always represents a valid regular polygon.

Most regular polygon problems start from a known side length and a sketch that names the shape. The calculator matches that workflow: the user picks n, types the side, and reads the result. The (1/2) * apothem * perimeter form still applies for back-solve from the apothem or circumradius.

When the project mixes regular polygons with rectangles, trapezoids, and other shapes, Area Calculator keeps the area work flowing on a single page instead of switching back and forth between shape-specific tools.

Three numerical factors drive the output: the number of sides, the side length, and the precision of pi and the trig functions.

• • The result is a decimal approximation, not an exact symbolic form. For exact symbolic results, derive them directly from the formulas.
• • The formulas assume equilateral and equiangular sides. For irregular polygons, polygons with a reflex angle, or polygons with holes, use a coordinate-based method like the shoelace formula instead.

The interior and exterior angle formulas are the easiest to verify: alpha = (n - 2) * 180 / n and beta = 360 / n, and they always sum to 180 degrees. These formulas hold for every regular n-gon with n >= 3.

According to Wolfram MathWorld: Regular Polygon, a regular polygon has equal side lengths and equal interior angles, and its area can be written as (n / 2) * R^2 * sin(2 * pi / n), where R is the circumradius and n is the number of sides.

According to Wikipedia: Regular polygon, a regular polygon is both equiangular and equilateral, and each interior angle of a regular n-gon measures (n - 2) * 180 / n degrees, with the polygon divisible into n congruent isosceles triangles meeting at the center.

If the polygon is not perfectly regular, the formulas on this page no longer apply, and Irregular Polygon Area Calculator handles the general case using the shoelace formula on a list of vertex coordinates.