# Math Help - Common area of circles

1. ## Common area of circles

Two circles have the same radius 1 unit. If each circle passes through the centre of the other circle, what is the area common to the two circles?

Area of two circles is 2pi. But how could I find the common area?

2. You'll need to make a sketch.

You'll see that the common area can be divided into 6 parts of which there are 2 equilateral triangles and 4 equal segments.

Can you try out to find the area now?

3. Originally Posted by Unknown008
You'll need to make a sketch.

You'll see that the common area can be divided into 6 parts of which there are 2 equilateral triangles and 4 equal segments.

Can you try out to find the area now?

So area of two equilateral triangles is $\frac {\sqrt 3}{2}.$

Area of 4 segment = $4 \times \frac{1}{2} (\frac {\pi}{3} - \frac{\sqrt 3}{2}) = \frac {2}{3} \pi - \sqrt 3$

Therefore, enclosed area = $\frac {\sqrt 3}{2} + \frac {2}{3} \pi - \sqrt 3 = \frac {2}{3} \pi -\frac {\sqrt 3}{2}$

Is this right?

4. Originally Posted by geton

So area of two equilateral triangles is $\frac {\sqrt 3}{2}.$

Area of 4 segment = $4 \times \frac{1}{2} (\frac {\pi}{3} - \frac{\sqrt 3}{2}) = \frac {2}{3} \pi - \sqrt 3$

Therefore, enclosed area = $\frac {\sqrt 3}{2} + \frac {2}{3} \pi - \sqrt 3 = \frac {2}{3} \pi -\frac {\sqrt 3}{2}$

Is this right? Yes
...

5. Yup, well done