There is a constant ratio between corresponding sides of similar figures. since circles don't have sides, what ratio(s) could you use to prove that all circles are similar?

A circle's size and shape is fully defined by its radius. Given two circles with radii r and r', the diameters are d=2r and d'=2r' and they are in the ratio 

d'/d = (2r')/(2r) = r'/r. 

The diameter ratio is the same as the radius ratio. Similarly, the circumferences c=πd and c' = πd' are in the ratio 

c'/c = (πd')/(πd) = d'/d = r'/r 

The circumference ratio is the same as the diameter ratio and the radius ratio. All of the key linear dimensions are in the same proportion. 

You might point out that the same thing happens with a square, where the size and shape are also completely determined by a single measurement, the length s of a side, with the diagonal and perimeter (corresponding to diameter and circumference) being d = √2 s and p = 4s. 

Maybe you can lead at least some of the students to generalize to other regular polygons. Some of them (like the equilateral triangle and regular hexagon) can be demonstrated like the square and circle above with formulas from geometry. The general case needs trig ratios to state the formulas relating side length to the radius and apothem of the polygon.