2026 AMC 8 Problem 11: Solution

Work through 2026 AMC 8 Problem 11 on arc length with a written solution, hints, and a visual explanation.

GeometryLevel 3 · AMC-style

Squares of side length 1, 1, 2, 3, and 5 are arranged to form the rectangle shown below. A curve is drawn by inscribing a quarter circle in each square and joining the quarter circles in order, from shortest to longest. What is the length of the curve?

A rectangle tiled with squares of side lengths 1, 1, 2, 3, and 5. A connected spiral consists of one quarter-circle arc inside each square.
Diagram 1
  • A. 4π
  • B. 6π
  • C. 13π/2
  • D. 8π
  • E. 10π
Hints
  1. Each quarter circle's radius equals the side length of the square it's inscribed in.
  2. Recall the arc length formula for a quarter circle: one-fourth of the full circumference, 2πr.
  3. Since every quarter arc shares the same factor of π/2, factor that out and just add the five radii before multiplying.
Read the step-by-step solution

Answer: B · 6π

  1. Find each arc length

    Each quarter circle has arc length (1/4)(2πr) = πr/2. The radii are 1, 1, 2, 3, and 5.

  2. Sum the arcs

    Total = π/2 × (1 + 1 + 2 + 3 + 5) = π/2 × 12 = 6π.

    π/2 × 12 = 6π

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