Circle Calculator
Give the circle calculator any one of radius, diameter, circumference or area and it returns the rest, exact in terms of π. It also does arcs, sectors and chords, the equation of a circle both ways, and the circle through three points.
Area
- Radius Given.
- 6
- Diameter d = 2r = 2 × 6 = 12
- 12
- Circumference C = 2πr = 2π × 6 = 12π. Equivalently πd.
- 12π ≈ 37.699112
- Area A = πr² = π × 6² = 36π. Equivalently πd²/4. In square units of whatever the radius was measured in.
- 36π ≈ 113.09734
- Double the radius Twice the size is twice the circumference and four times the area, because area grows with the square of length.
- circumference 75.398224, area 452.38934
| π taken as | Circumference | Area | Error |
|---|---|---|---|
| 3 | 36 | 108 | -4.5% |
| 3.14 | 37.68 | 113.04 | -0.051% |
| 22/7 | 37.714286 | 113.14286 | +0.04% |
| 3.1416 | 37.6992 | 113.0976 | +0.00023% |
| π (exact) | 37.699112 | 113.09734 | none |
- Exact forms in π are worked from the number as typed, so 2.5 is exactly five halves. The decimals are those exact values, rounded to eight significant figures.
- For the area of other shapes, use the area calculator. For a sphere, cylinder or cone, use the volume calculator.
How the Circle Calculator works
The two formulas everyone needs are C = 2πr and A = πr². Circumference is two pi times the radius (or pi times the diameter, which is the same thing), and area is pi times the radius squared. Take a circle of radius 6: the diameter is 12, the circumference is 2π × 6 = 12π, about 37.699, and the area is π × 6² = 36π, about 113.097. That is the worked example above.
Pi is the ratio of a circle's circumference to its diameter, and it is the same for every circle, which is why one constant does all the work. It is also irrational, so every decimal of it is a rounding. 3.14 is low by about 0.05%, and 22/7 is high by about 0.04%. Either is good enough for a garden bed but not for an exam that asks for an exact answer. That is why this calculator keeps answers in terms of π where it can: 36π is exact and 113.097 is not, and a textbook answer key usually wants the first. The table under the result shows what each common approximation of π does to your numbers.
Working backwards takes one rearrangement. From the diameter, halve it. From the circumference, r = C / (2π): π moves into the denominator, so a circumference of 10 gives a radius of 5/π, about 1.5915. From the area, r = √(A / π): an area of 36 gives a radius of 6/√π, about 3.385. Notice the difference between an area of 36 and an area of 36π. The second has a radius of exactly 6; the first does not, and misreading one for the other is the commonest slip in these problems.
Two other mistakes account for most wrong answers. The first is using the diameter where the formula wants the radius, which makes an area four times too big. The second is the units: area is in square units, so a circle 10 cm across has an area of about 78.54 square centimeters, not centimeters.
Scaling is worth having a feel for. Double the radius and the circumference doubles, but the area quadruples, because area grows with the square of length. One 18-inch pizza has 81π square inches; two 12-inch pizzas have 72π between them. The single large one is about 12.5% more food.
The other modes take the circle further. An arc is a piece of the circumference and a sector is a slice of the area, each the same fraction of the whole that the central angle is of 360°; the chord, the segment and the sagitta (the height of the arc above its chord) bring in sine and cosine. The equation of a circle is the distance formula in disguise: the points at distance r from the center (h, k) are exactly those with (x - h)² + (y - k)² = r². And three points that are not on one line fix exactly one circle, whose center is where the perpendicular bisectors of the chords between them cross: the circumcenter of the triangle they make.
For the area of other shapes, use the area calculator; for spheres, cylinders and cones, the volume calculator. Decimals here are shown to eight significant figures.
Circle Calculator FAQ
What is the formula for the circumference of a circle?
C = 2πr, or equivalently C = πd, since the diameter is twice the radius. A circle of radius 6 has a circumference of 12π, about 37.699.
How do I find the radius from the area?
Divide the area by π and take the square root: r = √(A / π). An area of 50 gives √(50 / π), about 3.989. If the area is given as a multiple of π, such as 49π, the radius is √49 = 7.
How do I find the radius from the circumference?
Divide by 2π: r = C / (2π). A circumference of 31.4 gives a radius of about 4.997. It is not exactly 5, because 31.4 is 10 times 3.14, not 10 times π.
Why leave the answer in terms of π?
Because it is exact. 36π is the area of a circle of radius 6; 113.097 is a rounding of it, and any further arithmetic on the rounding carries the error along. Convert to a decimal at the end, when you need a measurement.
What is the difference between a sector and a segment?
A sector is the slice bounded by two radii and an arc, like a slice of pizza. A segment is the region between a chord and its arc: the sector with its triangle cut away. So segment area = sector area - triangle area, and the arc mode above computes both.
Area, circumference, arcs and equations
- Area of a circle calculator
- Circumference calculator
- Equation of a circle calculator
- Arc length calculator
- Sector area calculator
- Chord length calculator