Maths Olympiad Prep

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Geometry Difficulty 3.3 AMC 10/12 Find the answer

Circles of radius 22 and 33 are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region:

Pick one

Solution

A line going through the centers of the two smaller circles also goes through the diameter. The length of this line within the circle is 3+3+2+2=10.3+3+2+2=10. Because this is the length of the larger circle's diameter, the length of its radius is 5.5.
The area of the large circle is 25π25\pi, and the area of the two smaller circles is 9π+4π=13π.9\pi + 4\pi = 13\pi. To find the area of the shaded region, subtract the area of the two smaller circles from the area of the large circle. ⟶25π−13π=(E) 12π\longrightarrow 25\pi - 13\pi = \boxed{\mathrm{(E) \ } 12\pi}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.