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# Geometry

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Two sectors of a circle of radius 12 overlap as shown, with P and R as the centers of the respective circles. Determine the area of the shaded region. Jun 22, 2022

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Let the intersection point directly above $$P$$ as $$A$$, and let the point that is to the left of $$P$$ be $$B$$

The area of $$\triangle ABP = 12 \times 12 \div 2 = 72$$

The area of the circular region $$ABP$$ is $${1 \over 4} \times 12^2 \times \pi = 36 \pi$$

Now, note that the area of the shaded region is $$2 (\text{circular region of ABP} - \triangle ABP)$$

Substituting what we know, we find that the area of the shaded region is $$\color{brown}\boxed{72 \pi - 144}$$

Jun 23, 2022

#1
0

Let the intersection point directly above $$P$$ as $$A$$, and let the point that is to the left of $$P$$ be $$B$$

The area of $$\triangle ABP = 12 \times 12 \div 2 = 72$$

The area of the circular region $$ABP$$ is $${1 \over 4} \times 12^2 \times \pi = 36 \pi$$

Now, note that the area of the shaded region is $$2 (\text{circular region of ABP} - \triangle ABP)$$

Substituting what we know, we find that the area of the shaded region is $$\color{brown}\boxed{72 \pi - 144}$$

BuilderBoi Jun 23, 2022