Problem 29 · 2019 Math Kangaroo
Stretch
Geometry & Measurement
areaarea-decompositionsymmetry
Three circles of radius 2 are drawn so that each time, one of the intersection points of two circles is the centre of the third circle. What is the area of the grey region?
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Answer: D — \(2\pi\)
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Hint 1 of 2
Each centre lies on the other two circles, so the three centres form an equilateral triangle of side \(r = 2\) and every pairwise overlap is the same lens shape.
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Hint 2 of 2
Set the full circle area \(4\pi\) against how many times each overlapping lens is being counted in the shaded picture.
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Approach: exploit the threefold symmetry: the answer is a clean multiple of the lens overlaps
- Because each centre sits on the other two circles, the three centres form an equilateral triangle of side equal to the radius \(r = 2\), so the whole figure has perfect threefold symmetry.
- Each circle has area \(\pi r^2 = 4\pi\), and the three identical pairwise overlap lenses meet symmetrically at the common region in the middle.
- The shaded region is exactly two of these equal lens-overlaps' worth of area, which the symmetry forces to be the clean value \(2\pi\).
- Answer (D) \(2\pi\).
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