Problem 1 · AMC 8 Stretch
Stretch
Geometry & Measurement
visual-representationseeking-complements
Square \(ABCD\) has side length \(1\). From two opposite corners, draw two quarter circles of radius \(1\). They overlap in a leaf-shaped (lens) region in the middle. Find the area of that shaded lens.
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Answer: \(\frac{\pi}{2}-1\) (about 0.57)
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Hint 1 of 4
Each quarter circle covers part of the square. What is the area of one quarter circle of radius \(1\)? (A full circle of radius \(1\) has area \(\pi\), so a quarter is \(\pi/4\).)
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Hint 2 of 4
Here is the key trick. Add the two quarter-circle areas together. Every point of the square gets covered, but the middle lens is the only part that sits inside BOTH quarter circles, so it gets counted twice.
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Hint 3 of 4
If you add the two quarter circles and then subtract the whole square once, the parts counted once cancel out and you are left with exactly the extra (doubly-counted) lens.
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Approach: Seeking complements — add the two quarter circles, then subtract the square once
- Each quarter circle has area \(\frac{1}{4}\pi(1)^2 = \frac{\pi}{4}\). Together their areas add to \(\frac{\pi}{4}+\frac{\pi}{4} = \frac{\pi}{2}\).
- When you lay both pieces on the square, they cover the whole square, but the leaf in the middle is covered TWICE (it belongs to both quarter circles). So the total \(\frac{\pi}{2}\) counts the square once plus the lens one extra time: \(\frac{\pi}{2} = (\text{square}) + (\text{lens}) = 1 + (\text{lens})\).
- Therefore the lens area is \(\frac{\pi}{2} - 1 \approx 0.57\).
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