8.6 SAT Math - Shaded Regions

Shaded region questions look much more difficult than they are.  The trick to shaded region questions is to look for a smaller shape within the larger.

-Hint: Though you may be tempted to divided the shaded region into smaller known shapes, it is almost always easier to subtract a smaller shape from a larger than to add up a bunch of small shapes.  Take a look at this one:

18.  If the radius of the circle above is 10, what is the area of the shaded region?
(A) 40 - 20π
(B) 100 - 25π
(C) 400 - 100π
(D) 100
(E) 400 - 20π

Sometimes shaded region questions will go a step farther and ask you about probability or the like.  You still need to solve the question in the same way, but your final answer may consist of a ratio or a little bit extra work.  Use the diagram below for the next two questions:

17. If the radius of the larger circle is 8 and the diameter of the smaller circle is 3, what is the area of the shaded region?

To solve, simply find the area of the smaller circle (π(1.5)²) and subtract it from the area of the larger circle (π8²).  You should get 64π - 2.25π or 61.75π.

18. If the radius of the larger circle is 8 and the diameter of the smaller circle is 3, what is the probability that a point selected will fall within the shaded region?

This question reads harder, but is very similar to the first one.  Since any point selected will fall within the area of the circle, find the areas of the larger circle and the smaller circle.  Subtract the smaller from the larger to get the area of the shaded region.  Then set up the probability ratio you already know, that is .  The desired outcome is the area of the shaded region, and the possible outcome is the area of the entire larger circle, so your ratio would be .