How the circle formulas work
π (pi) is the ratio of a circle's circumference to its diameter — an exact constant of about 3.14159265 that holds for every circle, at any size. Three formulas cover almost every circle problem:
Area A = πr² · Circumference C = 2πr · Diameter d = 2r
Because area scales with r² while circumference scales with r, doubling the radius quadruples the area but only doubles the perimeter — the reason pizza slices get edge-to-edge toppings in ever-larger circles. If you know the circumference, r = C ÷ 2π; if you know the area, r = √(A ÷ π).
Worked example
A circle of radius 5 has area π × 25 ≈ 78.54 square units and circumference 2π × 5 ≈ 31.42 units. Scale it to a sphere of the same radius (a ball of radius 5) and the volume is ⅔⁄₃πr³ = 523.60 cubic units. A cylinder with that radius and height equal to the diameter (h = 10) holds πr²h = 785.4 cubic units — exactly 1.5× the sphere, which is a good sanity check to remember.
Where circle maths gets practical
Pipe and tank capacity, circular cut-outs in sheet metal, arcs and fillets in machining, pizza and pie areas, sprinkler and irrigation coverage, the turning radius of a vehicle (wheelbase ÷ 2 × tan(steer angle)), and the arc length of a bend all start here. For a full annulus (a ring, or a pipe wall), subtract: A = π(R² − r²).