Problem 24 · 1992 AJHSME
Stretch
Geometry & Measurement
area-subtraction
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Answer: A — 7.7.
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Hint 1 of 3
The little shaded star sits inside the square but outside all four circles. So which big shape, minus which circular bites, leaves exactly that sliver?
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Hint 2 of 3
Each circle is centered at a CORNER of the square, where the angle is 90° — one quarter of a full turn. So each circle pokes in only a quarter. Four corner-quarters together make one whole circle.
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Hint 3 of 3
First nail the square's side: neighboring circles just touch, so the distance between adjacent centers is two radii.
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Approach: shaded = square − four corner quarter-circles (which total one whole circle)
- The four centers form a square. Adjacent circles touch, so the side equals two radii: 2 × 3 = 6, and the square's area is 62 = 36.
- At each corner the square's 90° angle captures exactly one quarter of that circle. Four quarters make one full circle of radius 3: area π · 32 = 9π ≈ 28.3.
- Shaded = square − the four corner-quarters = 36 − 28.3 ≈ 7.7.
- Why this transfers: a circle centered at a polygon corner always contributes a wedge equal to (corner angle ÷ 360°) of the circle — a quarter at a square corner. So four square-corner quarters conveniently reassemble into exactly one whole circle, sparing you from adding wedges separately.
- Sanity check: the circles fill most of the square, so only a thin star is left — an answer under 8 (not 17 or 27) is the believable one.
Mark:
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