How to find the area of a sector - GRE Quantitative Reasoning

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Question

Shadedwithtriangle

The circle above has a center of A, and points B and C lie on the circumference.

What is the area of the shaded region?

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Answer

We need to begin by finding the area of the following sector:

Sectorabc

If ∠BAC = 120°, then the area of sector BAC is equal to 120 / 360 = 1/3 of the entire circle. Since AC is 12 and is a radius, we know the total area is pi *122 = 144*pi. The sector is then 144*pi / 3 = 48*pi.

Now, we need to find the area of the triangle ABC. Since AB and AC are equal (both being radii of our circle), we have an icoseles triangle. If we drop a height from ∠BAC, we have the following triangle ABC:

12icotriang

We can use the 30-60-90 rule to find the height and base. One half of the base (x) is found by the following ratio:

√(3) / 2 = x / 12

Solving for x, we get: x = 6 * √(3). Therefore, the base is 12 * √(3).

To find the height (y), we use the following ratio:

1 / 2 = y / 12

Solving for y, we get y = 6.

Therefore, the area of the triangle = 0.5 * 6 * 12 * √(3) = 3 * 12 * √(3) = 36 * √(3)

The area of the shaded region is then 48*pi - 36 * √(3).

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