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What is the area of the shaded region?

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What is the area of the shaded region?

by BTGmoderatorLU » Thu Mar 29, 2018 1:47 pm
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A circular region centered at C is inscribed in equilateral triangle ABC above. If the area of the triangle ABC is 12, what is the area of the shaded region?

A. 6-3Ï€
B. 6-π√3
C. 12-π√3
D. 12-3Ï€
E. 3+π√3

The OA is B.

Area of equilateral triangle = √3/4 * a* a = 12.

Area of sector = 1/2 * Theta * (Ï€/180)*r*r

Here r = height of triangle = √3/2 * a

Area of the shaded region = 1/2 * (12 - Area of sector) = 6 - π√3.

Please, can anyone suggest another way to solve this PS question? Thanks in advance.
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by Jay@ManhattanReview » Thu Mar 29, 2018 11:26 pm
BTGmoderatorLU wrote:Image

A circular region centered at C is inscribed in equilateral triangle ABC above. If the area of the triangle ABC is 12, what is the area of the shaded region?

A. 6-3Ï€
B. 6-π√3
C. 12-π√3
D. 12-3Ï€
E. 3+π√3

The OA is B.

Area of equilateral triangle = √3/4 * a* a = 12.

Area of sector = 1/2 * Theta * (Ï€/180)*r*r

Here r = height of triangle = √3/2 * a

Area of the shaded region = 1/2 * (12 - Area of sector) = 6 - π√3.

Please, can anyone suggest another way to solve this PS question? Thanks in advance.
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by GMATGuruNY » Fri Mar 30, 2018 2:48 am
BTGmoderatorLU wrote:Image

A circular region centered at C is inscribed in equilateral triangle ABC above. If the area of the triangle ABC is 12, what is the area of the shaded region?

A. 6-3Ï€
B. 6-π√3
C. 12-π√3
D. 12-3Ï€
E. 3+π√3
Shaded region = (triangle area - sector area)/2 = (12 - sector area)/2 = 6 - (sector area)/2.
The expression in blue implies that the correct answer must be A or B.
Option A yields a negative value:
6 - 3π ≈ 6 - 9 = -3.

The correct answer is B.
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