one small circle and one big circle

Geometry Level 3

In figure shown above, the red line is 36 units long and it is tangent to the small circle. The diameter of the smaller circle is in line with the diameter of the big circle. Find the area of the shaded part.

1296 \pi

324 \pi

648 \pi

72 \pi^2

2 solutions

Edwin Gray
Jun 23, 2018

Let R = radius of large circle, r = radius of small circle.Area of shaded part - pi R^2 - pi r^2. Erecting a triangle at the center of the large circle, we have a right triangle with legs = r and 18, with hypotenuse = R. so r^2 + 324 + R^2, and pi (R^2 - r^2) = 324 pi. Ed Gray

A Former Brilliant Member
Apr 5, 2018

The area of the shaded region is equal to the area of the big circle minus the area of the small circle. We can transfer the small circle to the center as shown. By pythagoream theorem,

$R^2=r^2+18^2=r^2+324$ $\implies$ $R^2-r^2 = 324$

Then,

$\text{[shaded area]} = \pi (R^2-r^2)=\pi \times 324 = \boxed{324 \pi}$

one small circle and one big circle

2 solutions

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