Problem 19 · 2019 Math Kangaroo
Hard
Geometry & Measurement
perimetersymmetry
The diagram consists of three circles of equal radius R. The centres of those circles lie on a common straight line, where the middle circle passes through the centres of the other two circles (see diagram). How big is the perimeter of the figure?
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Answer: A — \(\dfrac{10\pi R}{3}\)
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Hint 1 of 3
Adjacent centres are a distance R apart, so neighbouring circles meet at 60° points.
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Hint 2 of 3
Add up the arc of each circle that lies on the outside of the figure.
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Hint 3 of 3
Two equal circles with centres R apart overlap in a 120° lens; work out each circle's exposed arc.
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Approach: add the exposed arcs of the three circles
- Neighbouring circles (centres R apart) cross at points 60° above and below the centre line.
- Adding the outside arcs of all three circles totals 600° of arc, i.e. 5/3 of a full circle.
- So the perimeter is (5/3)(2πR) = 10πR/3.
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