Try this beautiful problem from Geometry based on Area of Region in a Circle.

Area of Region in a Circle – AMC-10A, 2011- Problem 18


Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$. What is the area inside Circle $C$ but outside circle $A$ and circle $B$ ?

  • \(\pi\)
  • \(\frac{3\pi}{2}\)
  • \(2\)
  • \(6\)
  • \(\frac{5\pi}{2}\)

Key Concepts


Geometry

Circle

Rectangle

Check the Answer


But try the problem first…

Answer: \(2\)

Source
Suggested Reading

AMC-10A (2011) Problem 18

Pre College Mathematics

Try with Hints


First hint

We have to find out the area of the shaded region .Given that three circles with radius \(1\) and Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$.so if we draw a rectangle as shown in given below then we can find out the required region by the area of half of $C$ plus the area of the rectangle minus the area of the two sectors created by $A$ and $B$

can you finish the problem……..

Second Hint

Now area of the rectangle is \(2\times 1=2\)

Area of the half circle with center (gray shaded region)=\(\frac{\pi (1)^2}{2}\)

The area of the two sectors created by $A$ and $B$(blue region)=\(\frac{2\pi(1)^2}{4}\)

can you finish the problem……..

Final Step

Therefore, the required region (gray region)=area of half of $C$ plus the area of the rectangle minus the area of the two sectors created by $A$ and $B$=\(\frac{\pi (1)^2}{2}\)+\(2\times 1=2\)-\(\frac{2\pi(1)^2}{4}\)=\(2\)

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