The formula
Every property of a circle follows from its radius r and the constant π (≈ 3.14159):
diameter: d = 2r
circumference: C = 2πr
area: A = πr²
If you start from something other than the radius, just rearrange: r = d ÷ 2 from a diameter, r = C ÷ (2π) from a circumference, and r = √(A ÷ π) from an area. Once you have the radius, the three formulas above give everything else.
Worked example
For a circle with radius 5:
Diameter: 2 × 5 = 10.
Circumference: 2 × π × 5 ≈ 31.4159.
Area: π × 5² = π × 25 ≈ 78.5398.
Check from area: √(78.5398 ÷ π) = 5, back to the radius.
Why π shows up everywhere
π is the fixed ratio of any circle's circumference to its diameter — divide the distance around by the distance across and you always get the same number, about 3.14159. That single relationship is why the circumference is 2πr and, through integration, why the area is πr². It means every circle, large or small, is governed by exactly the same constant.
Note: the units follow the input. If your radius is in centimetres, the circumference is in centimetres and the area is in square centimetres.