Problem 28 · 2025 Math Kangaroo
Stretch
Geometry & Measurement
perimeterarea-decomposition
In the picture we see a regular octagon with a side length of 1 cm. Eight circular arcs with a radius of 1 cm and with centres at the corners were drawn as shown. What is the perimeter of the dark area?
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Answer: B — \(\dfrac{2\pi}{3}\) cm
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Hint 1 of 3
By the pinwheel symmetry the dark central region is bounded by eight congruent arcs, all of radius 1.
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Hint 2 of 3
An arc length on a radius-1 circle equals its central angle in radians, so you only need the angle of one arc.
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Hint 3 of 3
Compare the octagon's interior angle (\(135^\circ\)) with the angles the radii to neighbouring arc-endpoints cut off.
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Approach: find one arc's central angle, then multiply by eight
- The figure's rotational symmetry makes the dark region's boundary eight congruent radius-1 arcs centred at the eight corners.
- Working out the angle subtended at a corner from the octagon's \(135^\circ\) interior angle gives \(15^\circ = \dfrac{\pi}{12}\) per arc.
- Total perimeter \(= 8 \cdot 1 \cdot \dfrac{\pi}{12} = \dfrac{2\pi}{3}\) cm, choice (B).
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