How this circle calculator works
Every measurement of a circle is locked to every other one. Once you know any single value (radius, diameter, circumference, or area), the other three are fixed, no exceptions, no extra information needed. So you pick the one you know, type it in, and we solve for the rest, showing the substitution at each step with your actual numbers so you can follow the working or copy it into homework.
Where the answer has a clean exact form, we show it symbolically too. A radius of 5 gives an area of exactly 25π, and 78.54 is just that number rounded to 4 significant figures. Teachers often want the π form; measuring tapes want the decimal. You get both.
The formulas
Here r is the radius (center to edge), d is the diameter (all the way across), C is the circumference (the distance around), and A is the area (the space inside). The reason π shows up everywhere is that π is, by definition, the ratio of any circle's circumference to its diameter: about 3.14159. That one fact, plus d = 2r, generates every formula above. The area formula comes from slicing the circle into thin wedges and rearranging them into a near rectangle of height r and width half the circumference, which is πr, giving πr × r = πr².
Worked example
A circle with radius 5:
Diameter: d = 2r = 2 × 5 = 10.
Circumference: C = 2πr = 2 × π × 5 = 10π = 31.42.
Area: A = πr² = π × 5² = π × 25 = 25π = 78.54.
And in reverse: given an area of 78.54, the radius is r = √(A/π) = √(78.54/π) = √25.00 = 5.000.
Common values: radius 1 to 10
Handy for checking homework or estimating without a calculator. All values are rounded to 4 significant figures and were computed by this page's own code.
| Radius (r) | Circumference (C = 2πr) | Area (A = πr²) |
|---|---|---|
| 1 | 6.283 | 3.142 |
| 2 | 12.57 | 12.57 |
| 3 | 18.85 | 28.27 |
| 4 | 25.13 | 50.27 |
| 5 | 31.42 | 78.54 |
| 6 | 37.70 | 113.1 |
| 7 | 43.98 | 153.9 |
| 8 | 50.27 | 201.1 |
| 9 | 56.55 | 254.5 |
| 10 | 62.83 | 314.2 |
Notice the fun coincidence at r = 2: the circumference and area are both 12.57. That is the only radius where they match numerically (2πr = πr² exactly when r = 2). Below it the circumference number is bigger; above it the area runs away, because area grows with the square of the radius.
The mistake that costs the most points
Mixing up radius and diameter. If a problem says "a circle 12 cm across" and you plug 12 into A = πr², you get an area four times too big, because "across" means diameter and the radius is 6. Before using any circle formula, ask yourself one question: is my number center-to-edge or edge-to-edge? This calculator sidesteps the trap by making you say which one you have.
Working with shapes beyond circles? We have the same step-by-step treatment for triangles on the triangle calculator and right triangle calculator, and for 3D on the sphere volume calculator (a sphere is what you get when a circle grows up).