Tip #20 to Get a Top ACT Math Score (page 2)
"Mmmm … donuts."
Homer Simpson, The Simpsons (Fox, 1992)
I lied in the previous section. There is one more important formula that you need to memorize for the ACT: To find the area of a shaded region, just subtract the area of the smaller figure from the area of the larger 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 area of the little guy from the area of the big guy. The area of the large circle = π(r2) = π (62) = π (36). And the area of the smaller circle = π (r2) = π (42) = π (16). So 36π – 20π = 16π is the area of the shaded region.
Correct answer: J
An easy way to remember this is, "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 ACT has one question using it.
- Openings for 3 square windows, each 2 feet on a side, were cut from a rectangular wall 7 feet by 10 feet. What is the area, in square feet, of the remaining portion of the wall?
- A square is circumscribed about a circle of 9-inch radius, as shown below. What is the area, in square inches, of the region that is inside the square but outside the circle?
- 81 – 9π
- 144 – 9π
- 324 – 18π
- 308 – 81π
- 324 – 81π
- In the figure below, the area of rectangle ABCD is 48 square units. What is the area of the shaded trapezoid, in square units?
- The larger square in the figure below has 8-inch sides and circumscribes the smaller square. If the larger square intersects the smaller square at the four midpoints of the sides of the larger square, what is the area, in square inches, of the shaded region?
- D When a picture is described and not shown, draw it. That helps you see what to do next, and it helps avoid careless error. The area of the remaining portion equals the area of the wall minus the area of the windows:
- K The word "circumscribed" throws some kids. But, in this book you'll learn all that you need for the ACT. When they throw a tough word at you that we have not discussed in this book, they'll define it in the question or you won't even need it. Here we don't need the word; you can cross it out. It's totally implicit in the diagram. "Circumscribed" just means "drawn around." So the square is drawn around the circle, which is obvious from the diagram anyway! So don't get intimidated. Stay confident and focused. So, the area around the circle equals the area of the square minus the area of the circle. "Mmmm … donuts!"
- C Some kids see this and say, "I don't know the formula for the area of a trapezoid, so I can't do it." It's true that you could solve this question with the trapezoid formula, but you can also use donuts. They give you a back door on purpose. Don't get intimidated. Stay with it and look for another way! The area of the shaded trapezoid equals the area of the rectangle minus the areas of the two triangles:
- H First, always label all the info from the question into the diagram. That helps you see what to do next and to avoid careless errors. The area of the shaded region equals the area of the big square minus the area of the small square. We can do that with some special right triangles or Pythagorean theorems (Skills 25 and 24), but we can also just calculate the area of a shaded triangle and multiply by 4.
Area wall – 3(area window) = area remaining portion
(7)(10) – 3(2)(2) = area remaining portion
70 – 12 = area remaining portion
58 = area remaining portion
Area square – area circle = area around circle (18)(18) – π(9)2 = 324 – 81(π)
Area trapezoid = area rectangle – area of triangles
Area trapezoid = 48 – 0.5(6)(4) – 0.5(3)(4)
Area trapezoid = 48 – 12 – 6 = 30
Area shaded = 4(area triangle)
Area shaded = 4(0.5(4)(4))
Area shaded = 32
Go to: Tip #21
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