How this instrument works
A regular polygon's defining property is that all n of its sides share one length, s, so the boundary is really just that same segment repeated n times end to end. Adding s to itself n times is exactly what multiplication means, which is why the perimeter collapses to the single product P = n × s — there is no derivation to unpack beyond the definition of a regular shape itself, and no rounding is introduced anywhere along the way.
That simplicity is worth noticing next to the polygon's own area, which needs the apothem and a tangent term, A = ns² ⁄ (4·tan(π⁄n)). Perimeter skips all of that: for any whole n and any side length s, n × s is a clean rational number, never an approximation forced by an irrational ratio. A circle's circumference, by contrast, always carries π no matter how the radius is chosen — a flat-sided shape trades a smooth curve for boundary lengths that arithmetic alone can settle.
Push n upward while holding the circumradius fixed instead of the side length, and the polygon's perimeter climbs toward the circle that contains it, 2πr, edging closer with every added side but never quite arriving. Archimedes used exactly this fact around 250 BCE, computing the perimeters of 96-sided polygons inscribed in and circumscribed around a circle to pin π between 223⁄71 and 22⁄7 — one of the earliest rigorous numerical bounds on the constant, built entirely from polygon perimeters rather than a single circle.
- Enter the side count into the Number of sides, n field — any whole number from a triangle (3) upward.
- Enter the length of one edge into the Side length field; the formula assumes every other edge matches it exactly.
- Read Perimeter for the total distance around the shape, reported in the same unit used for Side length.
- Raise n while holding Side length fixed to watch the boundary grow in direct proportion — twice the sides means exactly twice the perimeter.
Worked example — a regular nonagon
A stop-sign-shaped frame is welded from nine equal steel bars, each cut to exactly 3 units. Its perimeter follows straight from the formula: P = n × s = 9 × 3 = 27 units — one multiplication, with no need to measure the other eight bars once the first length and the side count are known.
Set a circle around that same nonagon and its circumference, C = 2πr, comes out irrational for any radius chosen. The nonagon's 27 stays exact by comparison: swap in any whole n and n × s remains a clean rational answer, the trade a flat-sided shape makes for having countable edges instead of one smooth curve.
Questions
What is the formula for the perimeter of a regular polygon?
P = n × s, where n is the number of sides and s is the length of one side. Because every side of a regular polygon is equal by definition, the sum s + s + … + s written n times over collapses into one multiplication — a nonagon with 3-unit sides, for instance, has perimeter 9 × 3 = 27.
Why doesn't this formula need π or trigonometry, unlike the area formula?
Perimeter is addition in disguise — n copies of the same length s — so no ratio like π or tan(π⁄n) ever enters. The companion area formula, A = ns² ⁄ (4·tan(π⁄n)), does need trigonometry, because area depends on the apothem, a distance set by the angle between sides, not just their count and length.
How is this different from the pentagon, hexagon, or octagon calculators on this site?
Those pages fix n at one specific value — 5, 6, 8, and so on — and report area and perimeter for that single shape. This calculator takes n itself as an input, so one formula covers every regular polygon: a 15-gon or a 40-gon works exactly the same as a nonagon, just by changing the Number of sides, n field.
Does this formula work for an irregular polygon?
No. P = n × s assumes every side is the same length, which is only true for a regular polygon. An irregular polygon needs each side measured and added on its own, since nothing forces the lengths to match — treating its longest side as s and multiplying by n will overstate or understate the true perimeter.
What happens to the perimeter as the number of sides grows very large?
Held to a fixed circumradius, a regular polygon's perimeter climbs toward the circumference of the circle that contains it, 2πr, edging closer with each extra side but never quite reaching it. Archimedes used this fact around 250 BCE, computing the perimeters of 96-sided polygons to bound π — one of the earliest rigorous numerical estimates of the constant on record.
What is the smallest number of sides this calculator accepts?
Three. A closed polygon needs at least three sides to enclose any area at all, so n = 3 is the practical floor. At that value the general formula reduces to the perimeter of an equilateral triangle, P = 3s, the simplest case the whole rule contains.