All this talk about the new SAT is interesting and all, but we’ve still got two years to live with the old one, so let’s get back to our regularly scheduled PWNing.

Here’s a minorly important circle fact that I find a lot of students don’t know: when a wheel is rolling—without slipping—it makes travels a distance of one circumference each time it makes one complete turn. The SAT doesn’t test this often, but it’s certainly in the realm of testable concepts.

I think one of the easiest ways to understand this is to picture a roll of tape. If you hold down the loose end, and roll the tape along a table until it’s made one complete revolution, how much tape will you have unrolled? Obviously, one circumference. And how far did the roll travel? It traveled the same distance as the amount of tape it left on the table—one circumference.

Does that make sense?

It does? OK, cool. Then let’s try a tricky question together:

  1. In the figure above, a wheel with center B and a radius of 12 cm is resting on a flat surface. A diameter is painted on the wheel. If the wheel begins to rotate in a clockwise direction and rolls along the surface without slipping, how far will B travel before the painted diameter is perpendicular to the surface for the first time?
    (A)2π cm
    (B)4π cm
    (C)8π cm
    (D)12π cm
    (E)16π cm

Awesome question, right? I know.

To solve it, first note that the wheel will have to rotate 30º just to get to the point where the painted diameter is parallel with the surface. That might be easier to see if you draw a line through B that’s parallel to to ground, like so:

Once the diameter is parallel to the surface, it’s going to have to rotate another 90º to make the diameter perpendicular to the ground. So, in total, the circle will have to rotate 120º.

As is true of many circle questions, then, this one really comes down to ratios. 120º is 1/3 of a full 360º circle. If a circle travels a full circumference when it makes a full revolution, then it will travel 1/3 of a circumference when it makes 1/3 of a revolution. So all we need to do is find the full circumference, and then find 1/3 of it.

distance traveled = 8π

Easy, right?

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