Every circle formula, and its reverse
- Diameter: d = 2r, and r = d / 2
- Circumference: C = 2πr = πd, and r = C / 2π
- Area: A = πr², and r = sqrt(A / π)
- Arc length: C × angle / 360, with the angle in degrees
- Sector area: A × angle / 360, which also equals half the radius times the arc length
- Chord: 2r · sin(angle / 2)
- Segment area: sector area minus (r² · sin angle) / 2
- Sphere: volume 4/3 πr³, surface 4πr²
The reverse cases are the ones people reach for and rarely remember. Given an area of 78.54 square inches, the radius is sqrt(78.54 / 3.14159), which is 5. Given 31.42 inches of circumference, the radius is 31.42 / 6.28318, also 5. This page accepts any of the four as the input for exactly that reason.
Angles here are degrees
The central angle box takes degrees: 90 for a quarter circle, 180 for a half, 360 for the whole. Radians are shown in the output as a cross-check, along with the angle written as a multiple of π, because that is the form most formula sheets use. If you work in radians the two headline formulas get simpler — arc length is just r × angle and sector area is half r&sq; times angle, with no division by 360 anywhere. That simplification is the whole reason radians exist.
The mistakes that actually cost material
The common one is squaring the diameter instead of the radius. Area is πr², not πd², and using the diameter by mistake overstates the area by a factor of four. If you are working from a pipe or tube size, which is quoted as a diameter, halve it before anything else.
The second is mixing a length with an area. A circumference of 31.42 inches and an area of 31.42 square inches are not the same circle and not the same kind of number — one is a distance around the rim, the other covers the face. Watch the unit that comes back: plain inches for the first, square inches for the second.
The third is assuming an ellipse behaves like a circle. It does not. An ellipse has area π × half the long axis × half the short axis, which is straightforward, but its perimeter has no exact closed formula at all and every published version is an approximation. If your shape is not truly round, measuring the perimeter with a tape beats calculating it.
Where the round numbers come from
π is what you get dividing any circle's circumference by its diameter — 3.14159265..., running forever without repeating. This calculator uses the full value your browser carries, roughly sixteen digits, which is far more than any tape measure justifies. If you are checking against a textbook that specified 3.14, expect the third digit of the answer to differ; that is the textbook rounding, not an error here. For a radius of 5, full precision gives 78.5398 square units and 31.4159 around, while 3.14 gives 78.5 and 31.4. Neither matters when your material comes in whole sheets, and both matter when you are boring a hole to fit.
Questions people ask
How do I get the radius if I only know the area?
Divide the area by pi and take the square root. An area of 78.54 square inches gives 78.54 / 3.14159 = 25, and the square root of 25 is 5 inches. Set the selector to Area and the calculator does this and then reports the diameter and circumference as well. This comes up whenever a spec gives you a cross-sectional area — a duct, a cable bundle, a drain — and you need to know what physical size will deliver it.
What is the difference between a sector and a segment?
A sector is the pie slice: two radii and the arc between them, with the point of the slice at the centre. A segment is what a straight cut across the circle leaves behind: the arc and the chord that joins its ends, with no centre point. The segment is always the smaller of the two for the same angle, since it is the sector with the central triangle removed. Segments turn up in real work more than people expect — the cross-section of liquid in a horizontal cylindrical tank is a segment, which is why a half-full tank gauge is not linear.
Does this handle ovals or ellipses?
No. An ellipse needs two measurements, the long and the short axis, and while its area is a clean formula, its perimeter is genuinely hard — there is no exact elementary expression for it, only approximations of varying quality. Running an oval through a circle calculator by averaging the two axes gives an area that is close and a perimeter that is not. If you need the distance around an oval, run a tape around it.
Why does the sphere volume option also show a cylinder?
Because the relationship between them is worth knowing and easy to use. A sphere occupies exactly two thirds of the smallest cylinder that contains it — a cylinder whose height equals the sphere's diameter. Archimedes considered it his best result and had it carved on his tomb. Practically, it means you can estimate a ball's volume by picturing the can it would just fit inside and taking two thirds of that, without touching a calculator.