Derive the formula for the circumference of a circle with radius r by following these steps.

✓ 1. Because all circles are similar, set up a proportion of the circumference to the diameter for the two circles shown in the diagram.
#&\frac{\sf C_1}{d_1} = \frac{\sf C_2}{d_2}#&

✓ 2. Suppose d₁ = 1. By definition, what is the circumference of the smaller circle? C₁ = 1 π 2π

✓ 3. Suppose the radius of the larger circle is r. What is its diameter?
d₂ = r/2 r 2r

✓ 4. If d₂ = 2r, then the similarity equation in step 1 becomes
#&\frac{\pi}{\sf 1} = \frac{\sf C_2}{2r}#&.

Solve this equation for C₂.
C₂ = πr π/r 2πr

✓ 5. Because the radius r of the larger circle could have been any number, you have shown that for any circle,

#&{\sf C} = 2\pi r.#&