Circle Calculator
Type the radius, diameter, circumference or area — whichever one you know — and the other three appear instantly, along with arcs, sectors, segments and chords.
The circle
Fill in whichever one you know. The other three follow.
This circle
A slice of it
Chord from its height
The reverse problem, and the one that comes up on site: you can measure how deep an arc dips but not its angle. This works out the rest.
Common angles on this circle
| Angle | Arc length | Sector area | Chord |
|---|
How to use this calculator
Step by step
- Type into whichever of the four boxes you actually know — radius, diameter, circumference or area. The other three fill themselves in, and the box you typed in is highlighted so you can see which one is driving.
- Set the unit and the number of decimal places once; every result below follows them.
- Drag the angle slider, or type an exact angle, to cut a slice out of the circle. The drawing updates with it.
- The sector is the pizza slice from the centre. The segment is the smaller piece between the straight chord and the curved arc. Both are listed separately because they are easy to confuse.
- Use Chord from its height when you can measure how far an arc bulges but not its angle — the usual situation on a building site or in a workshop.
- The table at the bottom lists the arc, sector area and chord at nine common angles for the circle you have entered.
Good to know
- The sagitta is the perpendicular distance from the middle of a chord out to the arc. It is the easiest arc measurement to take with a straight edge and a ruler.
- Arc length is r × θ only when θ is in radians. The calculator converts from degrees for you, and shows the radian value so you can check.
- Area scales with the square of the radius: double the radius and the area quadruples, while the circumference merely doubles.
- A semicircle is a 180° sector, and its segment area equals its sector area — the chord passes through the centre. That is a quick way to sanity-check a result.
- For a circular tank, the area here multiplied by the depth gives the volume, and every cubic metre is 1000 litres.
- π is irrational, so every circle result is a rounded decimal. Raise the decimal places if you are chaining this result into another calculation.
The maths behind it
- Circumference C = 2πr = πd
- Area A = πr² = πd² ÷ 4 = C² ÷ 4π The last form is handy when only the circumference is measurable.
- Arc length s = rθ (θ in radians) Or s = 2πr × (degrees ÷ 360).
- Sector area A = ½r²θ The slice from the centre, bounded by two radii and the arc.
- Chord c = 2r·sin(θ ÷ 2) The straight line joining the two ends of the arc.
- Segment area A = ½r²(θ − sin θ) The sector minus the triangle inside it.
- Sagitta h = r(1 − cos(θ ÷ 2)) How far the arc bulges past its chord.
- Chord from sagitta c = 2√( h(2r − h) ) The reverse problem, solved without knowing the angle.
What is the difference between a sector and a segment?
A sector is bounded by two radii and the arc between them — the pizza slice, with its point at the centre. A segment is bounded by a straight chord and the arc above it — the slice you get by cutting straight across, with no point at the centre. The segment is always the smaller of the two for the same angle, and the difference between them is exactly the triangle formed by the two radii and the chord.
Why does typing in one box change the others?
Because a circle is completely determined by its size. Once you know any one of the four measurements, the other three are fixed — there is nothing left to choose. The highlighted box shows which value the calculator is treating as yours; the rest are derived.
I measured the sagitta and the chord. Can I find the radius?
Yes, and it is a genuinely useful trick: r = (c² ÷ 8h) + (h ÷ 2), where c is the chord and h the sagitta. Enter that radius at the top and the calculator gives you the rest of the arc. This is how you work out the radius of a curve you cannot get to the centre of.
Do I need to work in radians?
Not here — type degrees and the conversion happens for you. But be aware the underlying formulas need radians, so if you copy them into a spreadsheet you must convert first, or every arc and sector answer will be wrong by a factor of about 57.