Bunuel wrote:

In the figure above, XYZW is a square with sides of length s. If YW is the arc of a circle with center X, which of the following is the area of the shaded region in terms of s?

(A) πs^2 – (s/2)^2

(B) s^2 – πs^2

(C) s^2 – πs^2/4

(D) 4s – πs

(E) 4s – πs/4

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yukidaruma , I think this way might be faster (and it is essentially what you figured out, condensed).

(Area of square) - (Area of

sector WXY) = area of shaded region

In these kinds of problems, almost always, the key is: "The sector is what fraction of the circle?"

Find that fraction, in this case, by using the sector's central angle and the 360° of a circle.

The central angle of this sector is the vertex of what we are told is a square. So the sector's central angle = 90°. Thus:

\(\frac{Part}{Whole}=\frac{SectorArea}{CircleArea}=\frac{90}{360}=\frac{1}{4}\)

The sector is \(\frac{1}{4}\) of the circle.* We need the circle's area divided by 4.

The circle with radius = \(s\) has area: \(πr^2 = πs^2\)

Sector area? \(\frac{πs^2}{4}\)

Area of square = \(s^2\)

(Area of square) - (Area of sector) = area of shaded region

\(s^2 - \frac{πs^2}{4}\)

Answer C

Hope it helps.

*This fraction can be used in a few ways. Example: to find arc length, which here would = 1/4 of circle's circumference; or the reverse, to find circumference from arc length (here, = arc length * 4).
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