Circle Calculator

Your kid's school project needs the area of a circle with a 7-inch radius and you don't want to look up π every time. Type the radius, get area, circumference, and diameter back. Or work backward — type circumference, derive everything else.

The calculator shows the formulas alongside the answers, which is more useful than people assume. If your kid asks 'why π squared,' the formula's right there: A = πr². The same goes for circumference (C = 2πr) and diameter (d = 2r).

Uses π to enough decimal places that rounding error won't bite you in any practical context — engineering, geometry homework, woodworking. If you need more than 15 digits of π, you're not doing the kind of work this tool is for.

The fundamental relationships are: diameter d = 2r (twice the radius); circumference C = 2πr = πd; area A = πr². The constant π (approximately 3.14159265) is the ratio of any circle's circumference to its diameter — it's irrational, meaning its decimal expansion never terminates and never repeats. The calculator uses π to enough decimal places that any practical calculation rounds correctly.

A worked example: a circular table with a 7-inch radius. Diameter is 2 × 7 = 14 inches. Circumference is 2 × π × 7 ≈ 43.98 inches (so the table edge is roughly 44 inches if you wanted to wrap a string around it). Area is π × 7² = π × 49 ≈ 153.94 square inches. Each formula is independent — knowing one of the four (radius, diameter, circumference, area) uniquely determines the other three. The calculator runs the conversion in any direction.

A useful real-world example: you're cutting a circle from a sheet of plywood and you know you want about 200 square inches of material. You need the radius to draw the circle. From A = πr², solve for r: r = √(A/π) = √(200/π) ≈ √63.66 ≈ 7.98 inches. So an 8-inch radius gets you slightly more than 200 square inches; a 7.5-inch radius gets you 176.7, which falls short. Working backward from area to find radius is exactly the kind of calculation that's easy to get wrong by hand and trivial here.

Some practical limits: the formulas assume a perfect mathematical circle. Real-world 'circles' (a CNC-cut disk, a printed roundel, a pipe cross-section) have manufacturing tolerances. For engineering applications where dimensional precision matters, your real area might differ from the formula by 0.1–1% depending on the manufacturing process. For decorative or homework use, the formula is exact for any practical purpose. If you need calculations involving partial circles (sectors, segments, arc length), those are different formulas; the calculator handles full circles only.