How this instrument works
This sheet solves one specific rearrangement of the oldest ratio in geometry: given the circumference C, it returns r = C ⁄ 2π directly, skipping the diameter entirely. The relationship is nothing more than C = 2πr solved for r, but which of the three circle quantities you already hold, and which one you need next, decides whether that particular division deserves its own instrument. A tape looped around a manhole cover, a grain silo, or a satellite dish hands you C; an area, a sector, or an arc-length formula downstream wants r, not d.
The 2π is not an arbitrary constant tacked on for convenience — it is the radian measure of one full turn. Lay the radius off along the circle's own edge and it takes exactly 2π such radius-lengths, no more and no fewer, to close the loop back on itself. That single geometric fact is the entire content of C = 2πr, and reading it backwards, as C ⁄ 2π, literally counts how many of those radius-lengths a given circumference contains, then hands back the length of just one.
At the boundary, a circumference of zero collapses the circle to a single point and returns a radius of zero — the division simply hits its floor, nothing degenerate to chase. A real circumference is never negative, so every valid entry here maps to one unique, non-negative radius; there is no second branch or sign ambiguity to track, unlike formulas built on a squared term.
- Enter the measured distance around the circle into the Circumference field.
- The sheet divides that figure by 2π and reports the outcome in the Radius field.
- Keep one length unit throughout — metres in gives metres out on the Radius reading.
- Carry that Radius straight into an area, sector, or arc-length formula without ever detouring through the diameter.
Worked example — 10π metres of coping stone
A mason has just set 31.41592653589793 metres of coping stone around the rim of a circular reflecting pool, a figure the surveyor's wheel recorded to the last millimetre. Entering 31.41592653589793 into Circumference returns a Radius of exactly 5, with nothing to round away, because 31.41592653589793 is 10π carried to double precision, and 10π ⁄ 2π reduces to 5 on the nose.
That radius of 5 is the number the next order actually needs: a tarp cut to cover the pool sizes by area, π × 5² ≈ 78.54 square metres, a formula that calls for r directly rather than the diameter a supplier's cut sheet might otherwise expect. Skipping the diameter step also skips a place for rounding to creep in — divide the raw circumference by 2π once, and every downstream formula inherits the same figure of 5 exactly.
Questions
How do you find the radius from the circumference?
Divide the circumference by 2π. The identity C = 2πr rearranges directly to r = C ⁄ 2π, so a circumference of 31.41592653589793, which is 10π, gives a radius of exactly 5 — no diameter calculation sits between the two figures.
How is this different from converting circumference to diameter?
Diameter is twice the radius, so C ⁄ π gives d while C ⁄ 2π gives r — the two results differ by a factor of exactly 2, and swapping them is the single most common slip with this formula. This sheet reports r on its own, with no diameter figure in between.
Why divide by 2π instead of just π?
Because C = 2πr, not C = πr — the factor of 2 is there since the radius is half the diameter, given C = πd and d = 2r. Dividing by π alone returns the diameter rather than the radius, roughly double the correct value, which is the most frequent hand-calculation error here.
Can the radius come out negative or bigger than the circumference?
Neither happens for a real circle. Circumference and radius are both zero or positive by definition, and since 2π is greater than 1, dividing by it always shrinks the number — the radius this sheet returns is always smaller than the circumference that produced it.
Where does the 2π actually come from?
It is the radian measure of a full turn: laying the radius along the circle's own arc takes exactly 2π copies to close the loop. That geometric fact is the entire content of C = 2πr, and this sheet's division simply counts how many radius-lengths a given circumference contains.
Which unit should the circumference be entered in?
Any consistent length unit works, since the relationship is a pure ratio: metres in yields metres out, inches in yields inches out. Keep that same unit through whatever calculation follows, such as an area or arc-length formula built on the radius this sheet returns.