Regular polygon area and interpretation
The apothem reaches a side perpendicularly and the circumradius reaches a vertex. Length results use the supplied side unit, area uses its square, and the interior angle is displayed in degrees.
To find the radius, circumference, and full disk area of a circle, use the Circle Calculator and follow its own input conventions.
The formula
Split the polygon into identical center-to-side triangles. Half a side and the central half-angle determine the apothem; half the perimeter times that apothem gives the total area.
Verified worked example
A regular hexagon with side length 2 has perimeter 12 and apothem √3. Its area is 6√3, approximately 10.392305 square units, and its circumradius is 2.
How to use this calculator
- Enter a whole side count of at least three.
- Enter the common side length in your chosen length unit.
- Read the area and perimeter, distinguishing center-to-side apothem from center-to-vertex radius.
Regular Polygon input conventions
Regular means both equal sides and equal interior angles. Side count alone does not make an arbitrary polygon regular. Connecting its center to every vertex divides it into identical isosceles triangles, which explains the apothem and area formulas. The apothem is perpendicular to a side, whereas the circumradius reaches a vertex.
Interpreting regular polygon area
The interior angle approaches 180 degrees as the number of sides increases. However, keeping the side length fixed also enlarges the perimeter and radius; it does not hold the shape inside a fixed circle. All reported lengths use your side-length unit and the area uses its square. This page does not accept a list of irregular vertices or test whether a measured outline is truly regular.
Assumptions & limitations
What this calculation assumes
- The polygon is convex, equal-sided, and equal-angled.
What to keep in mind
- Convex regular polygons only; no irregular outlines or star polygons.
Common questions
What is the apothem?
It is the shortest distance from the polygon’s center to a side. In a regular polygon it is also the radius of the inscribed circle.
Can I use three sides?
Yes. A regular three-sided polygon is an equilateral triangle. A side count below three cannot enclose a polygonal region.
Do equal side lengths alone make a polygon regular?
No. A regular polygon must also have equal interior angles. For example, an arbitrary rhombus has equal sides but does not follow the regular square formulas unless its angles are right angles.
Sources & further reading
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