Problem 14 · 2014 Math Kangaroo
Medium
Geometry & Measurement
area-fractionsymmetry
The shaded part of the regular octagon has an area of 3 cm². What is the area of the whole octagon?
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Answer: D — 12 cm²
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Hint 1 of 2
Relate the shaded triangle to the whole regular octagon by symmetry.
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Hint 2 of 2
The shaded triangle is a fixed fraction of the octagon's area.
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Approach: use the octagon's symmetry to size the shaded triangle as a fixed fraction
- Draw the four long diagonals through the centre; they cut the regular octagon into 8 equal triangles meeting at the centre.
- The shaded triangle covers exactly 2 of those 8 equal pieces, so it is one quarter of the whole octagon.
- Therefore the octagon's area is 4 × 3 = 12 cm².
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