Problem

A circle has radius \(r=5~\text{cm}\) and diameter \(d=10~\text{cm}\). Evaluate or transform \(\text{both}~\text{circumference}~\text{forms}\).

Circle with radius labeled 5 cm and diameter labeled 10 cm.

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What this problem is really about

The two circumference forms describe the same circle using different size measures, so the bridge between them is the diameter-radius relationship. We’ll replace diameter with two radii, simplify each expression independently, compare the results, and check that circumference keeps linear rather than square units.

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C = πr² = 25π cm², so the forms are not equivalent.

C = πd = 10π cm and C = 2πr = 10π cm, so the forms are equivalent.

The forms are equivalent because d = r/2.

Both forms give 10π cm, so they are not equivalent.

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A circle has radius \(r=5~\text{cm}\) and diameter \(d=10~\text{cm}\). Evaluate or transform \(\text{both}~\text{circumference}~\text{forms}\).

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