Problem 16 · 2009 Math Kangaroo
Hard
Geometry & Measurement
perimetersymmetry
An equilateral triangle with side length 3 and a circle with radius 1 have the same centre. What is the perimeter of the figure created when the two are put together?
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Answer: A — \(6+\pi\)
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Hint 1 of 2
Decide where the circle pokes out past the triangle and where the corners poke out past the circle.
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Hint 2 of 2
The outline is the three straight side-pieces plus the bulging circular arcs; add their lengths.
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Approach: trace the union’s boundary, mixing straight pieces and arcs
- The circle (radius 1) reaches past each side’s midpoint, replacing a chunk of each side with an outward arc, while the sharp corners stay.
- Adding the leftover straight pieces and the three equal arcs, the lengths combine to 6 + π.
- The perimeter is 6 + π.
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