Using the calculator
Type one measurement with its letter: r = 5 for the radius, d = 10 for the diameter, C = 31.4 for the circumference or A = 78.5 for the area. A number on its own is read as whatever "A plain number is" says, the radius to start with. Units are a setting, so 5 cm and 5 work the same and the answers carry the unit you pick.
Add a central angle, as in r = 5, θ = 60°, and you also get the arc length, the sector area, the chord and the segment. The angle is in degrees unless you write rad or use π, so π/3 and 60° are the same angle. "Answer" picks which result sits at the top. The rest are all listed under it, each with its own copy button.
You can work backwards too. Give the radius and an arc length (r = 4, s = 2π) and it finds the angle; give an arc length and an angle and it finds the radius.
Where the numbers allow, each answer comes in exact form beside the decimal. A radius of 5 gives a circumference of 10π, about 31.4159, and π can go in the input as well: A = 25π gives a radius of exactly 5.
Circle formulas
| To find | From r | From something else |
|---|---|---|
| Diameter d | d = 2r | d = C ÷ π |
| Circumference C | C = 2πr | C = πd, or C = 2√(πA) |
| Area A | A = πr² | A = πd²/4, or A = C²/4π |
| Radius r | r = d/2, r = C/2π, r = √(A/π) | |
| Arc length s | s = rθ | s = 2πr × degrees/360 |
| Sector area K | K = r²θ/2 | K = πr² × degrees/360, or K = rs/2 |
| Chord c | c = 2r sin(θ/2) | |
| Segment area | r²/2 × (θ − sin θ) | the sector less the triangle |
Worked examples
A wheel 70 cm across. The diameter is 70, so C = πd = 70π cm, about 219.91 cm. That's how far it rolls in one turn. Its area is π × 35² = 1225π cm², about 3848.45 cm².
A circular rug with an area of 10 m². The radius is √(10/π), about 1.7841 m, so it's about 3.5682 m across and its edge is about 11.2100 m long.
A pizza slice. A 30 cm pizza cut into eight has a radius of 15 and a central angle of 45° per slice. The crust is s = 15 × π/4 = 15π/4, about 11.78 cm, and the slice covers 225π/8, about 88.36 cm².
Why the answers come in π
π is the ratio of any circle's circumference to its diameter, about 3.14159. It never ends and never repeats, so any decimal answer is rounded somewhere. Writing 10π keeps the answer exact until the last step, and maths teachers ask for it for that reason.
When you start from the circumference or the area, π ends up in the denominator or under a root. A circle with C = 10 has r = 5/π. One with A = 50 has r = √(50/π), which simplifies to 5√(2/π). When the exact form would be longer than it is useful, the calculator gives only the decimal.
Archimedes and the 96-sided polygon
Around 250 BC Archimedes pinned π between two fractions without measuring a single circle. He drew regular polygons inside and outside a circle and worked out their perimeters, doubling the sides from six up to 96. The circle's circumference had to lie between the two.
His result was 223/71 < π < 22/7, or between 3.1408 and 3.1429. 22/7 is still the fraction many people learn first, and it's within 0.04% of π.
Questions
What's the difference between a sector and a segment?
A sector is a slice cut from the centre, like a slice of pizza. A segment is the piece between a chord and its arc, so it's the sector with the triangle from the centre taken away.
Can the angle be more than 180°?
Yes, up to 360°. For a reflex angle the segment is the big piece, so the calculator adds the triangle back instead of taking it away.
Why are there two units, cm and cm²?
Lengths (radius, diameter, circumference, arcs, chords) are in the unit you pick. Areas are in that unit squared.
How accurate are the decimals?
They're worked from π to double precision and rounded to the decimal places you set, four to start with. The exact forms aren't rounded at all.
Sources
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