2. 外语
  3. In the figure above, if the square inscribed in the circle has an area of 16, what is the area of the shaded region?

In the figure above, if the square inscribed in the circle has an area of 16, what is the area of the shaded region?

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问题
In the figure above, if the square inscribed in the circle has an area of 16, what is the area of the shaded region?
选项 A、2π-1
B、2π-4
C、4π-2
D、4π-4
E、8π-4
答案B
解析 It is clear from the figure that the area of the shaded region is 1/4 of the difference between the area of the circle and the area of the square. You are given that the area of the square is 16, so each side has length 4.
You can find the area of the circle if you know the radius of the circle. If you draw a diagonal of the square, as shown in the figure below, you can see that the diagonal is also a diameter of the circle.

Note that the diagonal divides the square into two isosceles right triangles with legs of length 4. By the Pythagorean theorem applied to one of the right triangles, the length of the diagonal is equal to, or 4. Thus the radius of the circle is r =, and the area of the circle is πr2=π(2)2=8π. Therefore the area of the shaded region is, or 2π-4. The correct answer is Choice B.
This explanation uses the following strategies.
Strategy 4: Translate from a Figure to an Arithmetic or Algebraic Representation
Strategy 6: Add to a Geometric Figure
Strategy 8: Search for a Mathematical Relationship
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