How this instrument works
The circumference is the distance once around a circle, and it is tied to the radius by the tidiest constant in mathematics: C = 2πr. The ratio holds for a coin and for a planet's equator alike — divide any circle's distance around by its diameter and you get π, 3.14159265…, every time. That universality is why a single input is enough: fix the radius and the other two lengths are already decided.
The relation runs both ways with no loss. Because C = 2πr is a straight proportion, dividing a measured circumference by 2π recovers the radius exactly — no approximation beyond the digits of π itself. This is genuinely useful in the field: wrap a tape around a tree trunk, a pipe, or a column, and you have measured the one length otherwise hidden inside the object. Foresters size trees this way; plumbers size pipes.
A detail worth noticing: the formula is linear. Doubling the radius exactly doubles the distance around. Add one metre of radius to a hula hoop or to the Earth itself, and the loop grows by the same 2π ≈ 6.28 metres in both cases — a classic result that surprises almost everyone the first time they meet it.
- Leave the dial on 'From radius' and enter your figure in the Radius field — any unit works, as long as you read the results in that same unit.
- Read Circumference and Diameter; Result repeats the circumference as the headline figure.
- To work backwards, switch to 'From circumference' and type the measured distance around into the Circumference field.
- In that mode, Result reports the radius, and Diameter gives the width you can check directly against a ruler or caliper.
Worked example — one metre of radius
A round café table is built with a radius of exactly 1 metre. Edging strip needed: C = 2π × 1 = 6.283185 m — order six and a third metres and there is a little spare. The diameter, d = 2 × 1 = 2 m, is the number that matters at the doorway: a 2.05 m opening clears it, barely.
The same numbers run in reverse. Wrap a tape around the finished table and read 6.283185 m; switch the sheet to 'From circumference' and it returns r = 6.283185 ⁄ 6.283185 = 1 m on the nose. The round trip costs nothing and catches a mis-measured radius immediately.
Questions
What is the formula for the circumference of a circle?
C = 2πr, where r is the radius — or equivalently C = πd using the diameter. The two forms are the same statement, since d = 2r. With r = 1 the formula gives 6.283185, which is 2π itself; multiply by any other radius to scale it up or down.
How do I find the radius from a known circumference?
Divide by 2π, which is 6.283185 to six decimals. A trunk taped at 3.1 m around has a radius of 3.1 ⁄ 6.283185 ≈ 0.493 m and a diameter near 0.99 m. This sheet's 'From circumference' mode does that division for you and shows the rearranged formula as it works.
Is circumference the same thing as perimeter?
Yes — circumference is simply the name reserved for the perimeter of a circle and of curved figures generally. Both mean the total distance around the boundary. For polygons you sum the sides; for a circle the sum closes into the single term 2πr.
How accurate is using 3.14 instead of the full value of π?
Good to about one part in 2,000 — an error of roughly 0.05%. On a 10 m loop that is about 5 mm: invisible for woodwork and fencing, but real for machining. This sheet carries π at full double precision, so the only rounding you see is in the displayed digits.
Which unit should I enter the radius in?
Any unit you like — the formula is a pure ratio, so the answer comes out in whatever unit goes in. A radius in inches yields a distance around in inches; centimetres yield centimetres. Just keep one unit throughout rather than mixing, say, a taped measurement in centimetres with a diameter check in inches.