A bicycle wheel
The Circle
Does a wheel work because it's round? There are shapes of constant width that roll perfectly well yet make terrible wheels. So where's the difference?
Representation layers
Tyre, rim, hub, and the spokes between them. The spokes hold the hub suspended dead centre. As the wheel turns, every point on the rim takes its turn touching the ground while the hub never gets closer or further away.
Switch layers — the scene stays put
What's really going on
What makes a wheel useful isn't its roundness — it's that its centre stays at a constant height. There's a shape called a Reuleaux triangle: constant width, rolls smoothly between two planes, a board resting on top of it never bobs. So it rolls just fine. But you can't build a wheel from it, because its centre rises and falls as it turns; mount it on an axle and the vehicle starts bouncing. That's the circle's privilege: the equal-distance condition holds the centre exactly r above the road at every instant, without exception. The axle rides level. So 'x² + y² = r²' isn't a formula — it's the reason a wheel works, written down.
Analytic form
x² + y² = r²
Every point at distance r from the centre. Any (x, y) satisfying the equation sits on the circle; anything else falls inside or outside.
Same concept, elsewhere
Memorising one example gets you nowhere. You need to recognise the concept independent of its context.
A steering wheel — wherever you grip it, your distance from the centre is the same, so the torque is too
Ripples spreading from a stone in water: the set of points the wave reaches in equal time
Your phone's 'within 5 km of me' search — it is literally solving a circle equation
A stake and a length of rope: the oldest way to draw a perfect circle in a garden