# Consider 3 equal circles of radius r within a given circle of radius R each to touch the other two and the given circle as shown in figure, then the area of shaded region is equal to ?

##### 2 Answers

We can form an expression for the area of the shaded region like so:

where

To find the area of this, we can draw a triangle by connecting the centres of the three smaller white circles. Since each circle has a radius of

We can thus say that the angle of the central region is the area of this triangle minus the three sectors of the circle. The height of the triangle is simply

The area of the three circle segments within this triangle are essentially the same area as half of one of the circles (due to having angles of

Finally, we can work out the area of the centre region to be

Thus going back to our original expression, the area of the shaded region is

#### Explanation:

Let's give the white circles a radius of

The centroid is the center of the big circle so that's the distance between the center of the big circle and the center of the little circle. We add a little radius of

The area we seek is the area of the big circle less the equilateral triangle and the remaining

We scale by