Problem 11 · 2026 AMC 8
Medium
Geometry & Measurement
arc-length
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Answer: B — 6Ο.
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Hint 1 of 2
The spiral isn't one weird curve — it's five separate quarter-circles, one per square. In a square of side s, the inscribed quarter-circle has radius s. What is a quarter of that circle's circumference?
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Hint 2 of 2
Each arc is ΒΌ × 2Οs = Οs/2 — the arc length is just proportional to the side. So factor out Ο/2 and add the five sides.
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Approach: each square gives a quarter-circle; arc length scales with the side
- Break the spiral into its five pieces: in each square of side s, the inscribed quarter-circle has radius s, so its arc is one-fourth of the full circumference: ΒΌ × 2Οs = Οs/2.
- Every arc is just (Ο/2) times its side, so factor that out and total the sides 1, 1, 2, 3, 5: total = (Ο/2)(1 + 1 + 2 + 3 + 5) = (Ο/2)(12) = 6Ο.
- Why this transfers: when a curve is built from circular arcs, handle each arc as (its fraction of a turn) × (2Ο × its radius), then add — you never need to draw the whole curve, just account for each arc's radius and how much of a full turn it sweeps.
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