A wheel of diameter 1m rolls one full turn — exactly how far does it travel?
Every circle, whatever its size, has the same circumference ÷ diameter — that single ratio is named π.
Experiment
Hands-on experiment
🔮 Predict first — a wheel of diameter 1m rolls one full turn. How far does it go?
🛞 Roll the wheel
Change the diameter and measure the distance. Watch (distance ÷ diameter).
📖 Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Wheels, jars, sun and moon — humanity was surrounded by circles from the start, needing to know how far a wheel rolls and how big a round field is.
Ancient measurers discovered something astonishing: for any circle, circumference divided by diameter always gives the same value (3.14…) — the circle's fingerprint beyond size: π.
With π in hand the circumference (πd) computed itself, and the idea of slicing the circle and rearranging conquered the area (πr²) — turning a curve into straight pieces, the very seed of integration.
Insight
Insights from the video
“π is the circle's fingerprint.”
A coin or a Ferris wheel: circumference ÷ diameter = π. A value unchanged across sizes — the most famous case of the ratio lab's 'relationship with the size erased'.
“The circle's area fell to 'cut and unroll'.”
Slice a pizza infinitely fine and zigzag the pieces together: a rectangle of width πr and height r appears. Cutting curves into straight bits — the idea that becomes integration two millennia later.
Misconception
Common misconceptions
π is 3.14.
3.14 is only π's first three digits. π = 3.14159265… never ends and never repeats — an irrational. Any finite decimal is an approximation, not π itself.
Double the diameter, double the area.
The circumference doubles, but the area quadruples (similarity ratio 2 → area ratio 4). Why one 18-inch pizza beats two 9-inch pizzas — the mathematics of ordering.
Formula
Writing it as math
What the wheel and the pizza revealed, in mathematical language.
Defining π
The same value for every circle — hence circumference = πd = 2πr. π is an unending irrational.
The area
The unrolled rectangle has width πr (half the circumference) and height r — their product is the circle's area.
Under scaling
Lengths scale by the ratio; areas by its square — similarity's law, verbatim on circles.
In Real Life
Where you meet it in real life
Pizza economics
One 18-inch pizza equals four 9-inch pizzas in area. The 'one large vs two smalls' debate ends instantly with πr².
Track starting lines
Outer lanes of a 400m track start further forward — larger curve radii mean 2πr more to run. About 7.6m of correction per lane.
Bike speedometers
Wheel diameter plus rotation count gives speed — one turn = πd. Car odometers run on the same principle.
Satellite orbits
A satellite orbiting at radius 20,000km travels 2πr ≈ 130,000km per lap. Space uses the same π as a bicycle wheel.
Try Yourself
Test yourself
Q1Roughly how long is the rim of a hula hoop 60cm across?Show answer ▾
πd = 3.14 × 60 ≈ 188cm — about 1.9m. Just knowing 'a bit more than 3 diameters' makes the estimate.
Q2Pizzas of radius 10cm and 20cm — how do their areas compare?Show answer ▾
π×20² ÷ (π×10²) = 400/100 = 4 times. Double the radius, quadruple the area — the area ratio is the similarity ratio squared.
Q3A circle's circumference is 31.4cm. Its area?Show answer ▾
2πr = 31.4 gives r = 5cm, so area = π×5² = 78.5cm². Recovering the radius from the circumference and feeding it to the area — the two formulas joined.
💡 Try answering yourself before revealing it — getting it wrong is where learning starts.
Watch
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Connection
Concepts connect
Previous concept
Similarity
The similarity eye — 'sizes differ, ratios don't' — is the key that found π.
← Similarity labLeads to next
Trigonometric Ratios
Where circles meet angles, trig ratios begin — on to the ratios drawn by points on a circle.
Go to the Trigonometric Ratios lab →Related
Labs worth exploring together
Related lab
Irrational Numbers
Why π never ends or repeats — the poster child of irrationals.
Go to the Irrational Numbers lab →Related lab
Similarity
All circles are similar to each other — which is why there is only one π.
Go to the Similarity lab →