One number fixes the entire circle
A circle has a single free measurement. Pin down the radius and the diameter, circumference and area are already decided, because each of them is the radius multiplied by a fixed number. That is the reason this page takes whichever figure you happen to have instead of demanding the radius.
Which figure you have depends on how you got it. A tape wrapped around a pipe gives circumference. Calipers across a bolt head give diameter. A land listing or a spec sheet for a round tabletop often quotes area and nothing else. Each of those routes back to a radius, and the rest follows.
Working back to the radius
| You measured | Radius comes from | Worth knowing |
|---|---|---|
| Radius | r | The hardest one to measure on a physical object, since you have to locate the centre first. |
| Diameter | r = d / 2 | Only correct across the widest point. A tape laid slightly off centre reads short every time, never long. |
| Circumference | r = C / 6.28318 | The most forgiving measurement to take. Errors shrink by a factor of about 6.28 on the way back to a radius. |
| Area | r = √(A / 3.14159) | The square root flattens error, so a 10 percent area error moves the radius by roughly 5 percent. |
Those last two rows are the reason the circumference route is preferred on shop floors. Wrap a flexible tape around a pipe, read 138 mm, divide by pi, and the diameter lands at 43.93 mm. Even if the tape reading is off by a whole millimetre, the diameter shifts by only 0.32 mm. Trying to eyeball the same pipe across its widest point with a rigid rule rarely does that well.
Arc, chord, sector, segment
Four terms describe the parts of a slice, and they get swapped around constantly. The drawing above shades one of them at a time, so set the angle and watch which shape lights up.
- Arc is the curved stretch of rim between the two ends of the slice. It is a length, so it comes out in your chosen unit.
- Chord is the straight line joining those same two ends. Always shorter than the arc, and equal to the diameter when the angle hits 180 degrees.
- Sector is the pizza slice: two radii and the arc between them. Its area is the whole circle scaled by the fraction of 360 degrees you took.
- Segment is what sits between the chord and the arc, the piece left over when you cut a sector along its chord. Below 180 degrees it is smaller than the sector, above 180 it is larger.
The segment formula is the odd one out. It reads ½r²(θ - sinθ) with theta in radians, not degrees, which trips people up when they paste it into a spreadsheet. In Excel, feed it RADIANS(90) rather than 90 or the sine term returns nonsense.
A worked example you already have an opinion about
Pizza sizes are quoted by diameter, and area is what you eat. Two ten inch pizzas sound like more food than one sixteen inch, and they are not.
r = 16 / 2 = 8 in
A = π × 8²A = 201.06 in²r = 10 / 2 = 5 in
A = π × 5² = 78.54 in²A × 2 = 157.08 in²The single sixteen inch pizza gives 28 percent more surface. Area grows with the square of the radius, so every doubling of width quadruples what you get. The same maths decides whether a wider irrigation sprinkler beats two narrow ones and why a slightly larger drain pipe clears so much more water.
How many decimals to keep
Pi is irrational, so the digits never stop. That fact matters far less than people expect. Five decimal places, 3.14159, sizes anything you are likely to build. NASA's Jet Propulsion Laboratory reports using fifteen digits for interplanetary navigation, and at that precision the error around a circle the size of Earth's orbit is smaller than a molecule.
This page computes with the browser's double precision constant, about sixteen digits, then rounds only for display. So the decimals dropdown changes what you read, never what was calculated. The real limit sits at the other end. If you measured a pipe to the nearest millimetre, an area printed to six decimals is false precision dressed up as accuracy. Round the output to match the tool you measured with, and note the unit while you are at it.
Where this calculator stops
- Circles only. An ellipse has two axes and no closed form for its perimeter. Working it out exactly takes an infinite series, which is why engineers reach for Ramanujan's approximation instead.
- Flat shapes only. Spheres, cylinders and cones share the pi constant but not these formulas. Those belong in the Volume Calculator and the Polyhedron Calculator.
- The unit is a label. Switching the dropdown from cm to in relabels the results, it does not convert between systems. Convert your input first with the Length Converter, then paste the number here.
- Zero and negative input return nothing. A circle with no radius has no geometry worth printing, so the fields blank out rather than showing rows of zeros.
- Floating point drift past fifteen digits. Set decimals to Full and a value like 37.699111843077517 shows the last digit or two wandering. That is the binary format, not the formula.
- No unit tracking on area input. When you solve from area, the number you type is read as square units of whatever the dropdown says. Nothing checks that you typed square metres and not square centimetres.
Checks worth running before you cut anything
- Confirm the measurement you started from. Solving from diameter when you actually measured radius doubles every result, and the numbers look perfectly reasonable while doing it.
- Compare circumference against roughly three times the diameter. Pi sits a shade over three, so if the ratio is nowhere near that, the input mode is wrong.
- Check the area against the square it fits inside. A circle fills about 78.5 percent of its bounding square, so a 10 cm circle covers around 78.5 cm², never 100.
- For a sector, add your angle to the leftover angle and confirm the two sector areas sum to the full circle. Anything else means the angle went in as radians.
- Add cutting allowance after the calculation, not before. A blade kerf or a seam allowance is a fixed length around the rim, so it belongs on the circumference figure.
For the single figure on its own, the Circumference Calculator and Area Calculator skip the extra output and give you one answer.
