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How do you calculate the area of the overlap in a venn diagram that has the centre of each circle touching the edge of the other circle?
Wiki User
∙ 11y ago
You're not really talking about a Venn diagram here, but you have a Geometry problem with a diagram that looks like a Venn diagram.
If this is not your question, please re-ask it.:-)
This is hard to explain without drawing a diagram! Assume the circles have equal radii, and say their centers are O and P, and they intersect at Q and R. Draw OP, OQ, OR and QR.
The area enclosed by OQ, OR and the minor arc QR of circle O is a sector. Triangles OQP and ORP are equilateral because every side is a radius of a circle. So, angle QOR is 120 degrees which is 1/3 of a circle.
The area of the sector is (1/3)of (pi)r2 = (pi)r2/3. Now if we subtract the area of triangle OQR, what is left is half of what we want. Triangles OQR and PQR are congruent, and their areas add up to the same as triangles OQP and ORP.
In other words, triangle OQR equals triangle OQP, which is an equilateral triangle. The height of an equilateral triangle is [sqrt(3)]/2 times the base ... proof left for student ... so the area is bh/2 = r[(sqrt2)/2](r)/2 = r2(sqrt2)/4
Subtracting, you get (pi)r2/3 - r2(sqrt2)/4 with a common factor of r2 ==> r2[pi/3-(sqrt2)/4]. Double this and you get 2r2[pi/3-(sqrt2)/4].
Hope that's right.
Wiki User
∙ 11y ago
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