Question

In the figure above, \( O \) and \( P \) are the centers of the two circles. If each circle has radius \( r \), what is the area of the shaded region?

• A. \( \frac{\sqrt{2}}{2} r^2 \)
• B. \( \frac{\sqrt{3}}{2} r^2 \)
• C. \( \sqrt{2} r^2 \)
• D. \( \sqrt{3} r^2 \)
• E. \( 2\sqrt{3} r^2 \)

Hint:

For circle intersection problems, symmetry often creates equilateral or isosceles triangles—look for them to simplify area calculations.

Answer 1

The Correct Option is D

Step 1: Geometry setup.
Both circles have radius \( r \). The line \( OP \) is drawn through the centers, dividing the shaded region into two equilateral triangles. Step 2: Triangle identification.
Each triangle has side length \( r \). The area of an equilateral triangle of side \( r \) is: \[ A = \frac{\sqrt{3}}{4} r^2. \] Step 3: Total shaded region.
There are 4 such small equilateral triangles, hence total area: \[ 4 \times \frac{\sqrt{3}}{4} r^2 = \sqrt{3} r^2. \] Step 4: Conclusion.
The area of the shaded region is: \[ \boxed{\sqrt{3} r^2} \]