Circle Relationships Explorer

Pre-Algebra · Axiom Academy

One number is hiding inside every circle you'll ever draw. Unroll it, then catch it holding steady no matter how big the circle gets. How far is it around a circle? A circle has no straight sides to measure, yet the distance around it — its circumference — is locked to the distance across it — its diameter — by a single fixed number. That number is , and it never depends on how big the circle is. Watch the rim of the circle unroll into a straight line. Then we lay copies of the diameter end to end along it: three whole diameters fit, plus a little leftover. That leftover is what makes just past 3 — about 3.14159 . That count never changes: the circumference is always 3.14159 diameters around — so C = d. Make the circle any size — the ratio won't budge Drag the radius from tiny to huge. The diameter and circumference shoot up together, but keep your eye on C ÷ d : it stays glued to 3.14159 . That stubborn constant is , and it's the same for a coin and for a planet. C and d change with the circle; their ratio does not. C ÷ d = 3.14159, always — which is why C = d, or C = 2 r. Now the space inside — and there's π once more A circle's area hides the same number. Cover the disk with squares whose side is the radius r — little r × r tiles. However big you make the circle, it always takes about 3.14159 of those tiles to fill it. So A ÷ r² is too. It takes 3.14159 of the r × r squares to fill the disk — so A = r². Same constant, second relationship.

This is the written version of the interactive lesson above. See the full Pre-Algebra course.