invariance of π

Question

Early on we are told that π = C/D where C is the circumference and D the diameter of a circle. We are told π is a constant; that it does not get larger or smaller if the size of the circle changes. But no one ever proves it. It gets skipped in elementary geometry, and then we never get back to it. Can you invent a proof that a) does not involve limits and is simple enough to explain to high school geometry students; b) does involve limits c) uses calculus d) is different from all of the above.