Problem 7 · 2022 Math Kangaroo
Medium
Geometry & Measurement
perimeter
Four circles with radius 1 intersect each other as seen in the diagram. What is the perimeter of the grey area?
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Answer: D — \(2\pi\)
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Hint 1 of 3
The grey boundary is made only of circular arcs, and every arc is part of a circle of radius 1.
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Hint 2 of 3
You don't need the individual arcs — just find the total angle they sweep through.
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Hint 3 of 3
By symmetry the grey arcs together turn through one full revolution, so their angles add to 360°.
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Approach: the boundary arcs all have radius 1, so add their angles instead of their lengths
- Every piece of the grey boundary is an arc of one of the radius-1 circles, so its length is just (its angle) × 1.
- Total boundary length therefore equals the sum of all the arc angles, in radians.
- By the symmetry of the four equal circles, the grey arcs together sweep through exactly one full turn, \(2\pi\) radians.
- So the perimeter equals \(2\pi\), which is choice D.
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