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Math Help - Common area of circles

  1. #1
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    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?
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  2. #2
    MHF Contributor Unknown008's Avatar
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    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?
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  3. #3
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    Quote Originally Posted by Unknown008 View Post
    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?
    Common area of circles-circles.png

    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?
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  4. #4
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    Quote Originally Posted by geton View Post
    Click image for larger version. 

Name:	Circles.PNG 
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ID:	19540

    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
    ...
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  5. #5
    MHF Contributor Unknown008's Avatar
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    Yup, well done
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