Problem 20 · 1998 AJHSME
Hard
Geometry & Measurement
foldingarea-to-side
Let PQRS be a square piece of paper. P is folded onto R, and then Q is folded onto S. The area of the resulting figure is 9 square inches. Find the perimeter of square PQRS.
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Answer: D — 24 inches.
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Hint 1 of 2
Don't picture the exact shape after folding — track only the AREA. Each fold lays the paper onto itself, so the visible area halves every time.
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Hint 2 of 2
Two folds means the final area is the whole square's area cut in half twice — that's one quarter. Set that quarter equal to 9 and work back to the side.
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Approach: each fold halves the area; two folds leave a quarter
- Folding a flat sheet onto itself doubles its thickness and halves its visible area. Fold P onto R: area halves. Fold Q onto S: it halves again. So the final figure is ¼ of the square.
- That quarter is 9 in²: ¼·s² = 9, so the full square has area s² = 36, and its side is s = 6.
- Perimeter = 4 × side = 4 × 6 = 24 inches.
- Why this transfers: when a problem folds paper but only asks about area, skip the geometry of the resulting shape — just multiply the original area by ½ for each fold. Then a quick √ takes area back to side length.
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