Circle Calculator
Radius, diameter, circumference and area are four names for one number. Enter whichever one you have and the other three follow — along with the area and arc length of a sector, for when the question turns out to be about a slice rather than the whole circle.
Any one measurement
Whichever one you have. The other three are the same circle described differently, so one number settles all four.
Centre to edge. Every other measurement here is this number multiplied by something fixed.
For the slice below: 90° is a quarter, 360° gives the whole circle back. Leave it alone if you only want the circle itself.
A circle of radius 5 is 10 across, 31.4159 around, and encloses 78.5398 square units. A 90° slice of it covers 19.635, with a curved edge 7.85398 long.
- Radius
- 5
- Diameter
- 10
- Circumference
- 31.4159
- Area
- 78.5398
- Sector area at 90°
- 19.635
- Arc length at 90°
- 7.85398
- Everything is derived from the radius. Whichever measurement you supply is inverted to a radius first, so the four figures can never drift apart.
- Lengths carry no units. Enter a radius in metres and the circumference is in metres and the area in square metres — the arithmetic never needs to know which.
- Figures are rounded to twelve significant digits rather than a fixed number of decimals, which is what keeps a circle a millionth of a unit across from reporting an area of zero.
One number under four names
Every measurement of a circle is the radius multiplied by something fixed. So this page does one thing before it does anything else: it works out the radius. Four inverse formulas in, one forward set out. The alternative — a grid of twelve direct conversions — is twelve things to keep in agreement, and the day one of them is corrected is the day the tool starts contradicting itself.
d = 2r • C = 2πr • A = πr² • sector = (θ ÷ 360) × A- r, d
- Radius and diameter — centre to edge, and all the way across
- C, A
- Circumference and area
- θ
- The sector’s central angle, in degrees
The first of those is not really a formula: π is C ÷ d. That is what the symbol names. What makes it worth a name is that the ratio is the same for every circle — scale a circle up and the way round and the way across grow by exactly the same factor, so their quotient never moves. Measure a wheel, a plate and the equator and you get one number.
The sector needs no trigonometry at all. A central angle covers θ ÷ 360 of a full turn, and it takes exactly that share of the area and exactly that share of the circumference with it. A 90° sector is a quarter of each; a 360° sector is the circle back again, which is why this page allows the full turn rather than treating it as an edge case.
Cut the circle into wedges and straighten it out
C = 2πr is a definition rearranged, so it needs no argument. A = πr² does, and the argument is short enough to follow in your head — which is odd, because almost no circle calculator prints it.
Cut the circle into a lot of thin wedges, like slices of a pie. Lay them out in a row, alternating point-up and point-down, so the slices interlock. What you get looks like a parallelogram with two slightly bumpy long edges — and the thinner you cut the slices, the straighter those edges become. In the limit it is a rectangle.
Now read off its two dimensions. Its height is the radius, because that is how long each wedge is from its point to its crust. Its widthis half the circumference: half the wedges lie crust-up and contribute their curved edge to the top of the rectangle, half lie crust-down and contribute theirs to the bottom, so each long edge is made of half the circle’s rim. Multiply them: r × ½(2πr) = πr².
That is the whole derivation. Archimedes made it rigorous in Measurement of a Circle, where Proposition 1 states that a circle’s area equals that of a right triangle with one leg the radius and the other the circumference — the same rearrangement, folded into a triangle instead of a rectangle, and proved by exhaustion roughly nineteen centuries before calculus made it a one-line integral.
A radius of 5, all the way round
Start with r = 5. The diameter is 2 × 5 = 10. The circumference is 2π × 5 = 31.4159. The area is π × 5² = π × 25 = 78.5398. Those four numbers are the same circle, and any one of them would have produced the other three: given the circumference instead, the radius comes back as 31.4159 ÷ 2π = 5; given the area, as √(78.5398 ÷ π) = 5.
Now take a 90° slice of it. Ninety degrees is 90 ÷ 360 = one quarter of a full turn, so the sector takes a quarter of the area — 78.5398 ÷ 4 = 19.6350 — and a quarter of the circumference as its curved edge: 31.4159 ÷ 4 = 7.8540. Set the central angle to 360° and both figures return to the whole circle; set it to 0° and both go to nothing, while the circle itself is untouched.
Doubling matters more than it looks. A radius of 10 has twice the circumference of a radius of 5 but four times the area: 314.159 against 78.5398. Area goes as the square of the radius, which is the entire reason a 16-inch pizza is not twice a 12-inch one, and why a pipe of twice the bore carries four times the water.
What this calculator assumes
- A true circle. An ellipse has two radii and a circumference with no closed form at all — it needs an elliptic integral. Nothing here applies to one.
- Lengths carry no units. Enter a radius in metres and the circumference is in metres and the area in square metres. The arithmetic never needs to know which, and cannot warn you if you mix them.
- Degrees, not radians. The central angle is in degrees, 0 through 360. A sector wider than a full turn is not a sector, so 361 is rejected rather than wrapped around.
- Twelve significant digits, not fixed decimals. A tool that spans a bearing race and a running track has no one sensible decimal place. Rounding to significant figures is what keeps a circle a millionth of a unit across from reporting an area of zero.
- π to double precision.The value used is the one built into the browser’s arithmetic, accurate to about 16 significant digits — some ten orders of magnitude finer than any physical circle you could measure.
Circle FAQ
Why is π the same number for every circle?
Because all circles are the same shape at different sizes. Scale a circle up by a factor of k and every length in the picture scales by k — the way round and the way across both grow by exactly the same factor, so their ratio does not move. That ratio is π. It is why π is a constant rather than a property of a particular circle, and why 'circumference divided by diameter' is a definition rather than a formula you have to remember. Measure a bicycle wheel, a dinner plate and the Earth's equator and you will get 3.14159… every time, to whatever accuracy your tape measure has.
Why is the area of a circle πr²?
Cut the circle into a large number of thin wedges and lay them out alternately point-up and point-down, so they interlock. The result is very nearly a rectangle, and the more wedges you cut the nearer it gets. Its height is the radius, because that is the length of each wedge from point to crust. Its width is half the circumference, because half the wedges put their curved edge along the top and half along the bottom. So the area is r × ½(2πr) = πr². Archimedes made this argument rigorous in Measurement of a Circle, roughly nineteen centuries before calculus gave it a shorter proof.
How do I find the radius if I only know the area?
Divide by π and take the square root: r = √(A ÷ π). An area of 78.5398 gives √(78.5398 ÷ 3.14159) = √25 = 5. It is worth noticing that the radius grows as the square root of the area, which is why a 16-inch pizza is not twice the pizza of a 12-inch one — it is about 1.8 times as much, because the areas go as the squares of the diameters. Select 'Area' as the known measurement above and this page does the inversion for you.
What is the difference between a sector and a segment?
A sector is the pizza slice: the region bounded by two radii and the arc between them, with its point at the centre. A segment is what you get when you cut straight across with a chord — the smaller piece between the chord and the arc, with no point at the centre. This calculator returns sectors, because a sector is a plain fraction of the circle and needs no trigonometry: a 90° sector is exactly a quarter of the area and a quarter of the circumference. A segment is a sector with a triangle subtracted, which is a different calculation.
How do I work out the area of a slice of pizza?
Take the whole pizza's area and multiply by the slice's share of a full turn. A 14-inch pizza has a radius of 7 and an area of π × 49 = 153.94 square inches. Cut into 8 slices, each slice spans 360 ÷ 8 = 45°, so each is 45 ÷ 360 = one eighth of the pizza: 19.24 square inches. Enter 7 as the radius and 45 as the central angle above and that is the sector area you get back.
Is the circumference the same thing as the perimeter?
Yes — circumference is simply the name the perimeter goes by when the shape is a circle. Both mean the distance once around the outside. The word survives for circles because the measurement predates the general term, and because it is useful to have one word that says 'and this one is curved'. The arc length this page returns is a piece of that same circumference: the fraction of it that the central angle covers.
Sources and review notes
- MacTutor History of Mathematics, University of St Andrews — “Pi through the ages”, including Archimedes’ Measurement of a Circle and the polygon bounds that first pinned π between 3¹⁰⁄₇₁ and 3¹⁄₇
- NIST Digital Library of Mathematical Functions — the reference definitions of the elementary functions and constants this engine relies on
None of this can go stale. The relationships between a circle’s four measurements are definitions and one constant; the sources above are cited for the history and the reference definitions, not for the arithmetic.