Problem 5 · 1992 AJHSME
Medium
Geometry & Measurement
area-subtraction
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Answer: E — 5.
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Hint 1 of 3
The shaded part is "what's left" after the hole is punched out. So which two areas do you compare, and which way around?
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Hint 2 of 3
Whenever a shape is carved out of a bigger shape, the leftover region = big area − carved-out area. This subtraction idea works for any hole.
Still stuck? Show hint 3 →
Hint 3 of 3
Watch the circle's measurement: diameter is 1, so the radius (which goes in the area formula) is only 1/2 — a tiny circle.
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Approach: shaded = rectangle − the circular hole
- The shaded region is the rectangle with a circle removed, so shaded = rectangle area − circle area. Rectangle = 2 × 3 = 6.
- The hole has diameter 1, so radius 1/2. Its area is π(1/2)2 = π/4 ≈ 0.8 — less than one square unit.
- Shaded ≈ 6 − 0.8 = 5.2, and the closest whole number is 5.
- Sanity check: the circle is small (it would fit inside a 1×1 square), so it can shave off only about ¾ of a unit — the leftover should be just under 6, and 5 is the only nearby choice.
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