Find the area of the shaded region

[fork]

In the figure, A, B and C are the centres of three equal circles, each of radius 1 cm. Find the area of the shaded region.

I feel stump with this question. Thanks

 

[Data]

First, notice that the problem has lots of symmetry, so you might be able to save some work at some point. Now, the distances \|AB\|, \ \|AC\|, \ \|BC\| are all 1cm (see why?). Can you construct any triangles that you can find the areas of? Can you construct any circular arcs that you can find the areas of?

 

[thepatientmental]

No problem! Finding the area of the shaded region can be tricky, but we can break it down into smaller steps to make it easier. First, we need to identify the shape of the shaded region. In this case, it is a triangle formed by the centers of the three equal circles. Next, we can find the side lengths of this triangle by drawing lines from each center to the opposite vertex. These lines will form three equilateral triangles with side lengths of 2 cm (1 cm radius + 1 cm radius).

To find the area of the shaded region, we can use the formula for the area of a triangle: 1/2 * base * height. The base of our triangle is 2 cm and the height can be found by drawing a line from the center of one circle to the midpoint of the opposite side, creating a right triangle with a height of 1 cm (half of the base).

So, the area of the shaded region is 1/2 * 2 cm * 1 cm = 1 cm^2. Therefore, the area of the shaded region is 1 cm^2. I hope this helps!