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Maths-

SAT

Easy

Question

In the figure above, the circle has center O. What is the area of the shaded region?

$\frac{8 π}{3}$
$\frac{1}{3} (12 \sqrt{3} - 4 π)$
$\frac{1}{3} (8 π - 8 \sqrt{3})$
$\frac{1}{3} (8 π - 12 \sqrt{3})$

The correct answer is: $1 third left parenthesis 8 pi minus 12 square root of 3 right parenthesis$

The triangle OAB is an equilateral triangle and the area is given by $fraction numerator square root of 3 over denominator 4 end fraction a squared$
The Triangle OAB is an equilateral triangle with side length of 4
The area of the sector is given by $theta over 360 pi r squared$
The area of the shaded region = Area of Minor Sector AB - Area of Triangle OAB
$60 over 360 pi 4 squared minus fraction numerator square root of 3 over denominator 4 end fraction 4 squared equals 1 over 6 pi left parenthesis 16 right parenthesis space minus space fraction numerator square root of 3 over denominator 4 end fraction left parenthesis 16 right parenthesis equals 8 over 3 pi minus space 4 square root of 3 equals 8 over 3 pi minus space fraction numerator 4 x 3 over denominator 3 end fraction square root of 3 equals 1 third left parenthesis 8 pi minus 12 square root of 3 right parenthesis$

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