How this calculation works
Use equal sides and angles, solve the stated known length and decompose the polygon into center triangles. Keep circumradius and apothem distinct; one is not a substitute for the other.
s = 2R sin(π/n); a = R cos(π/n); perimeter = ns; area = perimeter × apothem / 2A worked example
A regular hexagon with side 4 m has perimeter 24 m, circumradius 4 m and apothem 2√3 m. Its area is 24√3 ≈ 41.569219 m² and each interior angle is 120°.
Good to know
Regular convex polygons only, not arbitrary equal-side shapes, star polygons or irregular surveyed boundaries. The explanatory SVG is a fixed hexagon, independent of entered side count.
Reference: MathWorld: geometry definitions and formulas (opens in a new tab) ↗
A side circumradius and apothem are different lengths
The circumradius runs from the center to a vertex. The apothem runs perpendicularly to the middle of a side. Neither is generally the side length. The known-length selector therefore changes the interpretation of the one dimension you supply.
For a square, a side of 4 m gives area 16 m². A circumradius of 4 m instead gives side 4√2 m and area 32 m². Keeping the numeral four while changing its meaning describes a different square, not a unit conversion.
The circle calculator is useful when comparing a fixed circumradius. Increasing sides while holding side length fixed changes the overall size instead.
Center triangles connect perimeter to area
A regular convex polygon can be divided into equal triangles from the center. Their combined area is perimeter × apothem / 2. The default regular hexagon with side 4 m has perimeter 24 m and apothem 2√3 m, giving 24√3 ≈ 41.569219 m².
The default hexagon’s 120° interior angle is different from its 60° exterior turning angle. Turning through one exterior angle at each corner completes a full 360° turn. A shape with equal side lengths alone need not have all the symmetry required by this calculation.
What stays fixed determines the side-count comparison
The table holds the selected known length and its meaning fixed. With circumradius fixed, increasing side count approaches the surrounding circle. With side length fixed, the overall polygon can grow instead; it is not a fixed-size circle comparison.
The polygon from coordinates calculator handles a separate supplied boundary and does not infer regularity from one length. The circle calculator supplies a useful fixed-radius comparison. The SVG uses a fixed hexagon to explain symbols; it does not redraw your entered side count or claim to show a million separate edges.
A few questions, answered
Can an arbitrary equilateral polygon use these formulas?
No. The supported shape is regular and convex, with equal sides and equal angles arranged symmetrically about the center.
Can I supply an apothem instead of a side?
Yes. Choose its meaning explicitly before entering the known length.
Is the diagram generated from the entered side count?
No. The fixed hexagon explains the length definitions independently of your numbers.
Does changing the common unit convert my input?
No. Convert the numerical length if you want to describe the same polygon in another unit.