Use the radius

Any time a circle appears in a problem, the first thing to think about is the radius of the circle. It is the core piece of information about the circle, and it is the thing that is most commonly used to calculate or infer information about things that appear in the rest of the problem.

The radius might be given numerically, but sometimes it can be given in a problem without an actual numerical value attached to it. For example, this is a problem from the December 2020 ACT.

Here it is not necessary to calculate the value of r, but instead simply to understand the relationship between each circle’s radii. The first circle has a radius of r, the second circle has radius of 2r, and the third circle has a radius of 3r.

The perimeter function for a circle is 2πr. Therefore, the sum of the circumferences of the circles is 2πr + 2π(2r) + 2π(3r). This can be rewritten as 6(2πr). The question said that the circumference of the smallest circle is 10, and since the circumference of the first circle = 2πr, then 6(2πr) = 6 • 10 = 60. So the answer is G.