# Another Challenging Shaded Region Problem

We’ve written about shaded region problems before (here and here) and recently offered up a challenging problem (here) on this topic. Here’s another problem that will put your math brain to work. Give it a try!

In the diagram below, segment AB is tangent to the circle at Point A. The measure of ∠OBA is 30 degrees and the circumference of the circle is 8π units. What is the area, to the nearest tenth of a square unit, of the shaded region?

A.  4.8
B.  5.5
C.  5.8
D.  6.0
E.  6.1

# ACT Math: A Challenging Shaded Region Problem

Our latest book, 200 Challenging ACT Math Problems, is in the proofreading stage and should be ready for publication in a couple of weeks. In the meantime, here’s a taste of the type of problem you’ll find in the book. See if you can figure out the answer to this challenging shaded region problem (Scroll down for the solution).

The large square in the diagram below has an area of 64 square units. A circle is inscribed in the large square and then a smaller square is inscribed in the circle. What is the total area, rounded to the nearest tenth of a square unit, of the shaded regions?

A.  16.0
B.  17.1
C.  18.3
D.  32.0
E.  34.3

# SAT Math: More Shaded Region Problems

If you’ve read my recent post on Shaded Region problems and you’d like some more practice, try the problems below.  Remember, our strategy isn’t to try to find each of these areas directly, but rather to use the areas of the regions we know to help us determine the area of the shaded region that we want to find.  The relationship in these problems is:

$Area_{shaded}=Area_{whole}-Area_{unshaded}$

Good luck!  The answers appear at the bottom of this post.

1)  Two semicircles are drawn inside circle B.  These semicircles intersect at point B, which is the center of the larger circle.  The length of AC is 8.  What is the total area of the yellow shaded regions?

# SAT Math: Shaded Region Problems

One of the most iconic of SAT Math problem types is the “shaded region” problem.  In this type of problem you are asked to find the area of a shaded region that is part of some larger diagram.  The key to successfully solving these problems is not to focus on the area of some oddly shaped region, but rather to focus on finding the areas of the shapes you know.  In general, you want to utilize the following relationship to solve these problems:

$Area_{shaded}=Area_{whole}-Area_{unshaded}$

Let’s take a look at how this works by solving the following problem.

In rectangle ABCD the length of side AB is 12.  The two circles are congruent and tangent. Additionally, each circle is tangent to the rectangle at three points.  What is the total area of the shaded regions?

(A)  $\: 36-18\pi$
(B)  $\: 72-9\pi$
(C)  $\: 72-18\pi$
(D)  $\: 72-36\pi$
(E)  $\: 144-36\pi$