How this instrument works
Perimeter is the general word for the total length of any closed boundary — a triangle's three sides added together, a hexagon's six, a circle's one continuous curve. For a circle that boundary has a name of its own, circumference, and a fixed formula, C = 2πr: multiply the radius by 2π and the distance around falls out exactly, because π is defined as the ratio of a circle's circumference to its diameter.
The formula isn't asserted, it's a limit. Inscribe a regular polygon with n corners inside the circle and its perimeter works out to 2nr·sin(π/n) — n chords, each subtending an angle of 2π/n at the centre. Let n grow without bound: sin(π/n) behaves like π/n for a small angle, so the polygon's perimeter creeps toward 2nr·(π/n) = 2πr. Archimedes ran exactly this argument by hand with 96-sided polygons to pin π between 223/71 and 22/7, roughly two thousand years before calculus gave the limit a name.
Shrink the radius to zero and the perimeter shrinks with it, in exact proportion — a point has no boundary to speak of, and the formula agrees without needing a special case. Polygons can't make that claim: a shape needs at least three sides to close, so there's no polygon with a smaller perimeter than a triangle of the same scale. A circle carries no such minimum; it is the smooth limit every polygon is reaching for, not a member of the polygon family itself.
- Type your circle's radius into the Radius field, in any unit you like.
- The sheet applies C = 2πr immediately and writes the result into the Circumference field.
- Read Circumference as the full distance around the circle's curved boundary — its perimeter.
- Change the Radius value at any time; Circumference recalculates instantly, no reset required.
Worked example — edging a radius-5 circular bed
A circular garden bed has a radius of exactly 5 (say, feet). The edging needed for one full lap is C = 2π × 5 = 10π = 31.41592653589793 ft — call it 31.42 ft on the tape measure, with the extra digits there only to show the figure wasn't rounded before you saw it.
Compare that to a hexagon traced inside the same circle: six straight chords, each equal to the radius, giving a perimeter of 6 × 5 = 30 ft — about 4.5% short of the true 31.41592653589793 ft. Add more sides to the polygon and that gap keeps closing, which is the very limit the formula C = 2πr captures exactly.
Questions
What is the formula for a circle's perimeter?
C = 2πr, where r is the radius — equivalently C = πd using the diameter, since d = 2r. Plug r = 5 in and you get 10π = 31.41592653589793, the exact figure this sheet returns before any rounding for display.
Why is a circle's perimeter called circumference rather than just perimeter?
Perimeter (Greek peri-, 'around', plus metron, 'measure') names the boundary length of any closed shape. Circumference (Latin circum-, 'around', plus ferre, 'to carry') was reserved specifically for circles and other smooth curves, long before the two words settled into everyday use as near-synonyms for a circle's case.
How is C = 2πr actually derived, rather than just given?
It's the limit of a regular polygon's perimeter as the number of sides grows. An n-sided polygon inscribed in the circle has perimeter 2nr·sin(π/n); as n increases, that expression closes in on 2πr, because sin(π/n) approaches π/n for small angles. Archimedes used this exact idea with 96-gons to bound π by hand.
What's the most common mistake when computing this?
Swapping radius and diameter — plugging a measured diameter into the Radius field, which doubles the true perimeter, or halving a true radius by mistake, which understates it by half. Since the diameter is always twice the radius, checking which one you actually measured is worth doing before you trust the result.
Does C = 2πr still apply to a semicircle or a partial arc?
The curved part scales the same way: an arc spanning angle θ, measured in radians, has length rθ, so a full turn, θ = 2π, gives back C = 2πr exactly. A semicircle's curved edge alone is πr; add the straight diameter, 2r, if you need the total distance around the closed half-circle shape.
How much does an error in my radius affect the perimeter I get back?
By exactly the same percentage, no more and no less, because C is directly proportional to r. A radius that's 1% too large returns a perimeter that's 1% too large; there's no compounding or shrinking of the error, which is what 'linear' means for this formula in practical terms.