Tip #18 to Get a Top SAT Math Score

OK, I lied in the previous Skill. There is one important formula that is not provided in the info box. To find the area of a shaded region, just subtract the area of the smaller figure from the area of the bigger figure. You might also be asked to find the perimeter of a shaded region.

Let's look at this question:

Solution: The area of a shaded region is found by subtracting the little guy from the big guy. So 10 – 8 = 2 and 2 is the area of the shaded region.

Correct answer: 2

An easy way to remember it: "The area of a donut equals the area of the big guy minus the area of the donut hole." This is a great formula. Almost every SAT has one question using it.

Example Problems

Medium

1. In the figure above, the perimeter of square BCDF is 24, DE = 4, and AB = 12. What is the area of the shaded region?
   1. 204
   2. 180
   3. 144
   4. 36
   5. 24



2. The small circle above is tangent to the large circle. If the radius of the large circle is 8, what is the area of the unshaded region?
   1. 48π
   2. 45π
   3. 32π
   4. 16π
   5. 8π



3. In the figure above, what is the area of the shaded region?

Hard

4. In the figure above, triangles ADB and EDC are similar. If AD = 4, ED = 2, and BD = 6, what is the area of the shaded region?

Answers

1. C   If the perimeter of square BCDF is 24, each side is 6 and the area is 36. The area of ACEG is A = (l)(w) = (18)(10) = 180. So the area of the shaded region is 180 – 36 = 144.
2. A   The area of the larger circle is A = πr² = π(8)² = 64π. The radius of the smaller circle is half the radius of the larger circle. So the area of the smaller circle is A = πr² = π(4)² = 16π. The area of the unshaded region is 64π = 16π = 48π.
3. 39   Fun question. Remember that the area of a shaded region equals area of big guy minus area of donut hole. The area of the whole rectangle is 6 × 9 = 54, and the area of the unshaded triangle is (this is the cool part) a = 0.5bh = 0.5(5)(6) = 15. It looks like the height wasn't given, but it is; it's 6, it's as tall as the rectangle. So the shaded region = 54 – 15 = 39.
4. 9   Since the triangles are similar, their sides are proportional (more on this in Skill 25). Therefore, , so CD = 3. (How would you know to set this up? You'll see that Mantra 25 says, "Similar triangles have sides that are proportional," so when the SAT says "similar," that's what they want you to set up. That's how to think like a math genius!) The area of the larger triangle is 0.5(base)(height) = 0.5(4)(6) = 12, and the area of the smaller triangle is 0.5(base)(height) = 0.5(2)(3) = 3. So the area of the shaded region is 12 – 3 = 9.

Go to: Tip #19