First, I'd suggest Erathostenes method of computing the radius of the Earth, unless it is really well-known to them.
Otherwise, here's a nice one. You go for a bike ride. At the end, each wheel has made a closed simple curve, and, suppose, you noticed that these two curves never met. What's the area encosed between them? The distance between the centers of the wheels is $r$.
Answer : $\pi r^2.$ Explanation: the line passing for the two contact points of the wheels on the ground, is tangent to the back wheel's curve, and during the trip, it made a complete $2\pi$ rotation. To convince your audience: imagine a disk (a pizza) cut into several very thin slices (with vertex in the center as usual). Make the slices slide on each other, so that a hole appears in the middle of the pizza. The curve bounding the hole is the one traced by the back wheel; the exterior boundary is the curve traced by the other wheel.
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