Problem 16 · 2017 Math Kangaroo
Medium
Geometry & Measurement
area-decomposition
Raphael has three squares. The first has side 2 cm. The second has side 4 cm, and one of its corners sits at the centre of the first square. The third has side 6 cm, and one of its corners sits at the centre of the second square. What is the total area of the figure shown?
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Answer: A — 51 cm²
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Hint 1 of 2
Add the three square areas, then subtract the parts that overlap where a corner sits at a centre.
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Hint 2 of 2
Each overlap is a quarter of the smaller square in that pair.
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Approach: add areas and subtract the overlaps
- The squares have areas 2² = 4, 4² = 16 and 6² = 36, totalling 56.
- Each larger square has a corner at the previous square's centre, overlapping a quarter of the smaller square: 4÷4 = 1 and 16÷4 = 4.
- Subtract the double-counted overlaps: 56 − 1 − 4 = 51.
- The figure's area is 51 cm² (A).
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