Problem 10 · 2009 Math Kangaroo
Medium
Geometry & Measurement
areaarea-decomposition
The area of the triangle shown equals 80 m². Each circle has a radius of 2 m and its centre is at one of the vertices of the triangle. What is the area of the grey shaded region (in m²)?
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Answer: B — 80 − 2π
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Hint 1 of 2
The grey region is the triangle with three circular wedges removed at the corners.
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Hint 2 of 2
The three corner angles of any triangle add to 180° — a half turn.
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Approach: subtract the three corner sectors from the triangle
- At each vertex a circular sector of radius 2 is cut out of the triangle.
- The three sector angles are the triangle's interior angles, which total 180° = half a full circle.
- Together they form half a circle of radius 2: area = ½·π·2² = 2π.
- Grey area = 80 − 2π, which is 80 − 2π.
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