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Free Online Circle Calculator With Arc and Sector

Enter the radius, diameter, circumference or area of a circle and get all four, together with the arc length, chord, sector area, segment area and segment height for any angle. The segment uses the formula that stays accurate at small angles, where the usual subtraction of two nearly equal areas loses its digits.

Free Forever Nothing Uploaded Any Measurement In Arc And Sector

Free Online Circle Calculator With Arc and Sector

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⭕ Circle Calculator Any one measurement gives the rest · arc, chord, sector and segment

The sum runs in your browser. Nothing you type is sent anywhere.

⭕ What you know

Any one of the four fixes the circle completely. Choose whichever measurement you actually have rather than working out the radius first.

The measurement you have

Radius

Sector angle, in degrees From 0 to 360. This is the slice of the circle used for the arc, chord, sector and segment below.

Try one of these

The seventh is a sector so narrow that the usual segment formula gives zero.

The area

—

Radius—

Diameter—

Circumference—

Arc length—

Chord across the sector—

Sector area—

Segment area—

Segment height—

Recent circles, kept in this browser only.

Nothing here yet Press Log to history on the Calculate tab to keep a circle here.

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How to Use the Circle Calculator

Three steps, and the third is the useful one.

Choose which measurement you have Radius, diameter, circumference or area. Any one of them fixes the circle, so there is no need to work out the radius first and no chance of an error doing so.

Read the other three They appear together, so a circumference becomes an area without a second step. The exact value in terms of pi is given alongside the decimal where it is tidy.

Set a sector angle Any angle from 0 to 360 gives the arc length, the chord across it, the sector area, the segment area and the segment height. Those five are what circle questions actually ask for.

What to Know About Circle Measurements

Which measurement is which, and where the arithmetic is delicate.

The segment area is computed directly, not by subtraction. A segment is the part of a sector left over once the triangle to the centre is removed, and the obvious way to find it is to work out both and subtract. At small angles those two areas are nearly identical, so the subtraction throws away almost every significant digit. The single formula, half r squared times the angle minus its sine, gives the same number without that cancellation. For a sector of a hundredth of a degree the difference between the two methods is total.

The four measurements are interchangeable. Radius, diameter, circumference and area each determine the other three. Which one you have depends on what you measured: a wheel gives a diameter, a running track gives a circumference, a piece of material gives an area. Being made to convert to a radius first is where mistakes creep in, so all four are accepted directly.

Pi is irrational, so every answer here is approximate. No circle has both a rational radius and a rational area. The figures shown are correct to about fifteen significant digits, which is the limit of the arithmetic in any browser, and the exact form in terms of pi is given alongside where it is tidy enough to be useful.

Arc, chord, sector and segment are four different things. The arc is the curved edge, the chord is the straight line joining its two ends, the sector is the whole wedge from the centre, and the segment is only the part between the chord and the arc. They are often confused, so each is labelled and each has its formula in the working.

Degrees in, radians shown. The angle is entered in degrees because that is how most people think about a slice, but every circle formula is written in radians, and the conversion is the first line of the working. The radian value is shown so that the formulas can be followed.

It runs in your browser with nothing uploaded. The arithmetic is in the page, so it works offline once loaded and nothing you type is sent anywhere.

Key Features & Capabilities

What it does, and what it deliberately does not.

Any measurement in Radius, diameter, circumference or area, and the other three come straight out.

Arc, chord, sector and segment All four for any angle, with the segment height as well.

Accurate at small angles The segment formula avoids subtracting two nearly equal areas, which is where the usual method collapses.

The exact form in pi Given alongside the decimal wherever it is tidy, because that is the answer homework wants.

The working shown Each formula with your own numbers in it, starting with the conversion to radians.

Nothing is uploaded It runs in this tab and keeps working offline.

About Circles

A circle is fixed by a single number, so the radius, the diameter, the circumference and the area each determine all the others. Which one you happen to have depends on what you measured. A wheel gives you a diameter, a track gives you a circumference, a sheet of material gives you an area. Most calculators ask for the radius and leave the conversion to you, which is both unhelpful and the place where errors get introduced, so all four are accepted here.

Beyond the whole circle, the useful quantities come from a slice of it. The arc is the curved part of the edge, the chord is the straight line between the ends of that arc, the sector is the wedge running back to the centre, and the segment is just the sliver between the chord and the arc. The arc and the sector are simple fractions of the circumference and the area. The chord comes from trigonometry. The segment is the one with a subtlety in it.

The natural way to find a segment is to take the sector and subtract the triangle formed by the two radii and the chord. That is correct and, for a narrow sector, numerically hopeless: the sector and the triangle are nearly the same size, so the subtraction cancels most of the digits in both and leaves an answer built from whatever rounding error remains. For a sector of one hundredth of a degree the subtraction method returns zero.

Writing the segment as half the radius squared times the angle minus the sine of the angle gives exactly the same quantity in a single expression, and at small angles the difference between an angle and its sine is computed without catastrophic cancellation. That is the version used here, which is why a very narrow sector still gives a real area rather than zero.

One more thing worth saying plainly: pi is irrational, so no circle has a rational radius and a rational area at the same time. Every decimal on this page is an approximation, accurate to about fifteen significant digits, and the exact value in terms of pi is given alongside wherever it is short enough to be useful.

Frequently Asked Questions

Radius, sectors, segments and degrees against radians.

No. Radius, diameter, circumference and area are all accepted, and whichever one you give produces the other three. Converting first is unnecessary and is where mistakes happen.

A sector is the whole wedge from the centre, like a slice of pizza. A segment is only the part between the straight chord and the curved arc, so it is the slice with the pointed end cut off.

Because the usual method subtracts the triangle from the sector, and at small angles those two are nearly identical. The subtraction destroys the digits that carry the answer. The single formula used here computes the same value without that loss.

The box takes degrees, which is how most people describe a slice, but the radian value is shown in the working because that is what the formulas use. Multiply by 180 and divide by pi to convert.

Because pi is irrational and cannot be written exactly in decimal. The figures are correct to about fifteen significant digits, and the exact value in terms of pi is given alongside where it is tidy.

The greatest distance from the chord to the arc, measured at right angles to the chord. It is what you need when a segment has to be marked out physically, such as an arched opening.

No. The arithmetic runs in your browser, so nothing you type leaves the page and it works offline.