Regular Polygon Calculator
Area, perimeter, apothem and interior angle of a regular polygon.
Area
64.9519
6-gon, side 5
Perimeter
30
Interior angle
120°
Apothem
4.3301
AI Breakdown & Smart Takeaway
Plain-English insight on your numbers
Get a personalized explanation of what these results mean — and how to improve them.
How the Regular Polygon Calculator works
The Regular Polygon Calculator instantly computes the area, perimeter, apothem, and interior angle of any regular polygon — from a triangle to a 100-sided figure — given just the number of sides and one measurement. It's an essential tool for students, architects, designers, and engineers working with symmetric geometric shapes.
A regular polygon is defined by two properties: all its sides are equal in length and all its interior angles are identical. This perfect symmetry means that just two inputs — the number of sides (n) and a single linear measurement such as side length, circumradius, or apothem — are mathematically sufficient to derive every other property of the shape. The calculator uses these inputs to produce the full geometric profile of the polygon in one step, eliminating the need for manual trigonometry.
The area of a regular polygon is calculated using the relationship between the apothem (the perpendicular distance from the center to the midpoint of any side) and the perimeter. Specifically, Area = (1/2) × Perimeter × Apothem. Under the hood, the apothem itself is derived from the side length and the number of sides via the formula a = s / (2 × tan(π/n)), where s is the side length. For a hexagon (n = 6), this simplifies elegantly due to the 30-60-90 triangle geometry embedded in its structure. Understanding this layered relationship helps you see why scaling the side length has a squared effect on area but only a linear effect on perimeter.
Interior angles are another key output. Each interior angle of a regular polygon equals (n − 2) × 180° / n, a formula derived from the fact that any polygon can be divided into (n − 2) triangles. For a pentagon, this gives 108°; for a hexagon, 120°. A common mistake is confusing interior angles with central angles (which equal 360°/n) or with exterior angles (which equal 180° minus the interior angle). The calculator clearly distinguishes all of these, reducing errors in applications like tile cutting, woodworking, and CAD drafting where the wrong angle can ruin a project.
One practical subtlety is the choice of input measurement. The calculator supports side length, circumradius (radius of the circumscribed circle passing through all vertices), and apothem (radius of the inscribed circle tangent to all sides). Many real-world problems give you the circumradius — for example, when fitting a polygon inside a circular pipe or bore — while others give the side length directly. Using the wrong input type is one of the most frequent errors users make, so always confirm which measurement you are working with before entering a value.
Formula
Area = n·s² / (4·tan(π/n))
Pro tips
- If you know the circumradius (R) rather than the side length, convert it first using s = 2R × sin(π/n) before entering values, or use the calculator's circumradius input mode directly to avoid conversion errors.
- For tiling and flooring projects, remember that only regular polygons with 3, 4, or 6 sides (triangles, squares, hexagons) can tile a flat surface without gaps — use the interior angle output to verify this: only angles that divide evenly into 360° will tessellate.
- When designing structures with rotational symmetry, the central angle (360°/n) tells you the exact rotation increment needed between identical components — useful in CAD, laser cutting, and CNC routing.
- Doubling the side length of any regular polygon quadruples its area, since area scales with the square of the linear dimension — always re-run the calculator rather than mentally scaling area values.
- For large n values (say, n = 72 or more), the polygon closely approximates a circle; cross-check the area against π × apothem² to verify your inputs are reasonable, since the two values will be nearly identical.
Key terms
- Regular Polygon
- — A polygon in which all sides are equal in length and all interior angles are equal in measure.
- Apothem
- — The perpendicular distance from the center of a regular polygon to the midpoint of any one of its sides, equivalent to the inradius of the inscribed circle.
- Circumradius
- — The radius of the circle that passes through all vertices of a regular polygon, measured from the center to any corner.
- Interior Angle
- — The angle formed inside the polygon at each vertex, calculated as (n − 2) × 180° / n for a regular polygon with n sides.
- Perimeter
- — The total boundary length of the polygon, equal to the number of sides multiplied by the length of one side.
- Central Angle
- — The angle subtended at the center of a regular polygon between lines drawn to two adjacent vertices, equal to 360° divided by the number of sides.
