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?

Shaded Region 1

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

Let’s start by trying to figure out the area of the rectangle.  We know that the length of the rectangle is equal to twice the diameter of the circle.  The width of the rectangle is equivalent to the diameter of the circle, or half the length of the rectangle.  So now we know that the width is 6.  That makes the area of the rectangle 6 x 12 = 72.

Now let’s find the area of one of the circles.  We know that the diameter of the circle is 6, so the radius is 3.  The area of one of the circles is:

Area_{circle}=\pi r^{2}=\pi (3)^{2}=9\pi

Now we have enough to find the area of the shaded region:

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

Area_{shaded}=Area_{rectangle}-2\cdot Area_{circle}

Area_{shaded}=72-2\cdot 9\pi

Area_{shaded}=72-18\pi

The correct answer is choice (C).

For more practice on Shaded Region problems, go here.

If you have questions about the Shaded Region problems or anything else to do with the SAT, leave a comment or send me an email at info@cardinalec.com .

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