Problem 23 · 2026 AMC 8
Stretch
Geometry & Measurement
belt-around-circles
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Answer: C — 4Ο + 20.
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Hint 1 of 2
Split the band into two kinds of pieces: the straight stretches (lying along flat tangent lines between coins) and the curved stretches (hugging a coin). Handle the two kinds separately.
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Hint 2 of 2
Key fact: going once around any convex bunch, the band turns through exactly 360Β°, so all its curved pieces together make one full circle. The straight pieces are each tangent, so they're as long as the gaps between the outer coin centers — i.e. the perimeter of the polygon joining those centers.
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Approach: curves sum to one whole circle; straights trace the outer-center polygon
- Diameter 4 means radius 2. Break the band into straight tangent segments and curved arcs that wrap the outer coins.
- The arcs: as the band loops all the way around, its direction turns through a full 360Β°, and each arc bends along a radius-2 coin. All the arcs together sweep one complete turn, so they add up to exactly one full circle: 2Ο × 2 = 4Ο.
- The straights: each tangent segment runs parallel to the line joining two neighboring outer coin centers and has the same length. So the straight pieces total the perimeter of the trapezoid through the four outer centers. With touching radius-2 coins, the bottom span is 8 and the other three center-to-center sides are each 4: 8 + 4 + 4 + 4 = 20.
- Band length = 4Ο + 20 → 4Ο + 20 centimeters.
- Why this transfers: for a belt/band tight around any bunch of equal circles, the answer is always (one full circle from the radius) + (perimeter of the polygon through the outer centers). The curved parts re-assemble into a single circle because the total turning is one full 360Β°.
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