Circumference and Area of a Circle

1. 1Guess the RimThe rim is longer than the diameter — but how many diameters long? Three? More? Less?C = 15.7 cm?
2. 2Wrap a String Around ItWrap a string once around the rim. The gold line traces exactly one trip around the circle.One trip around the rim
3. 3Unroll the StringThe string comes off the rim and lies straight. Now measure it — with the diameter itself.The rim, laid straight
4. 4Three Diameters and a Bit MoreOne, two, three diameters land along the string one by one — and a short violet piece is still left over. The counter only climbs when a diameter really lands.3 diameters + a short piece
5. 5The CircumferenceSo the rim is the diameter times 3.14: C = 10 × 3.14 = 31.4 cm. The 3.14 grew out of the string we just counted.C = d × 3.14 = 31.4 cm
6. 6The Hinge: Cut, and the String ShrinksFor the area, cut the disc into 16 equal wedges — the cut lines appear one by one around the centre. At the same time the measured string shrinks to exactly half the rim: the C/2 segment, waiting to become the top edge of a strip.16 equal wedges · C/2 = 15.7 cm
7. 7The Wedges Fly Into a StripOne by one, left to right, the wedges fly into a strip, alternating point-up and point-down. The old disc stays behind, pale as a ghost — it has become the strip. The strip is half the circumference long, 15.7 cm, and one radius wide, 5 cm.Length C/2 = 15.7 cm · Width r = 5 cm
8. 8The AreaArea of the strip: 15.7 × 5 = 78.5 cm². And 15.7 is itself 5 × 3.14 — so S = r × r × 3.14 = 78.5 cm².S = r × r × 3.14 = 78.5 cm²
9. 9Check ItSweep across the wedges in the strip and compute directly: 5 × 5 × 3.14 = 78.5 ✓ — the same answer the strip gave.5 × 5 × 3.14 = 78.5
10. 10The Ghost and the StripThe ghost disc and the finished strip sit side by side: the same amount, two ways of seeing it. Circumference 31.4 cm; area 78.5 cm².C = 31.4 cm · S = 78.5 cm²

Change the numbers to test the engine with any sum & difference problem: