Wednesday, February 25, 2015

Standardized test prep -- Another circle problem

[Another SAT/GRE style geometry problem, slightly more advanced than the last one.]

The circle pictured here has radius 5 with a center indicated by the black dot. What is the area of the shaded region?

How to get started: You may have already thought of this own your own (it’s OK to get ahead of the class here), but if you’re stuck, it’s often a good idea  to ask yourself “is there a simpler related question I can answer?” In this case, how about finding the area of the whole circle?

Now we just need to figure out what part of the circle is shaded. Since the angle corresponding to the shaded region is vertical with the angle that measures x degrees, x degrees is also the measure of that angle. That means that the ratio of x to 360 is the same as the ratio of the area of the shaded region to the area of the circle.

How do we figure out what x is? Look at the two labeled angles. Put together they form a straight line. That means that the two angles are supplementary.

That gives us x equals thirty and since thirty is one twelfth of 360…

The area of the shaded region is one twelfth the area of the circle.


  1. I'd change the radius to 9 as having two 5's in the problem can be confusing to a learner. It also makes the fraction at the end reducible which teaches kids that it's good practice to look out for that.

    I'd also put in an extra step in equations to really spell it out e.g.
    area of a circle = pi * r^2 = pi * 9 * 9 = 81pi

    It's pitching it at the kids at the bottom but even kids in the middle can struggle over some really simple things.

    1. That's an excellent point about the fives and the nines.

      I already added the extra step you mentioned

      area of a circle = pi * r^2 = pi * 9 ^2 = 81pi

      to the video I'm working on.