Question:medium

Two identical circles intersect so that their centres, and the points at which they intersect, form a square of side 2 cm. What is the area of the portion that is common to both the circles?

Updated On: Jan 13, 2026
  • $2(\pi– 2) cm^2 $
  • $2(\pi – 1) cm^2 $
  • $2\pi cm^2 $
  • $(\pi – 2) cm^2 $
  • $(\pi – 1) cm^2 $
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The Correct Option is A

Solution and Explanation

The correct answer is option (A):
$2(\pi– 2) cm^2 $

Let's break down this geometry problem step by step to understand how to arrive at the correct answer.

We have two identical circles intersecting in such a way that their centers and the intersection points form a square. The side of this square is given as 2 cm. This immediately tells us the radius of each circle is also 2 cm. Why? Because the distance from the center of a circle to an intersection point is the radius. Since the side of the square is 2 cm, the distance from the center to an intersection point (which is along the side of the square) is 2 cm.

Now, consider the area common to both circles. This area is made up of two equal segments of the circles. To calculate the area of one segment, we can use the following approach:

1. Focus on one quarter of the shape: Consider one of the segments formed. Its boundaries are two radii (forming a right angle at the center since it is a square) and a circular arc.
2. Calculate the area of the sector: The sector is formed by the two radii and the arc. The angle formed at the center of the circle by these radii is 90 degrees (a quarter of the full 360 degrees). The area of a sector is given by (1/4) * pi * r^2. In our case, this would be (1/4) * pi * (2^2) = pi cm^2.
3. Calculate the area of the triangle: The triangle is formed by the two radii and the chord connecting the intersection points. This is a right-angled isosceles triangle with legs equal to the radius, 2 cm. The area of the triangle is (1/2) * base * height = (1/2) * 2 * 2 = 2 cm^2.
4. Calculate the area of the segment: The area of the segment is the area of the sector minus the area of the triangle: pi - 2 cm^2.
5. Calculate the total common area: Since the total common area is made of two equal segments, the common area is 2 * (pi - 2) = 2(pi - 2) cm^2.

Therefore, the area of the portion common to both circles is 2(pi - 2) cm^2.
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