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Fibonacci Spiral — Definition, Formula & Examples

A Fibonacci spiral is a curve built by drawing quarter-circle arcs inside squares whose side lengths follow the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, …). Each arc connects smoothly to the next, producing a spiral that closely approximates the golden spiral.

Given a tiling of rectangles constructed from squares with side lengths equal to consecutive Fibonacci numbers FnF_n, the Fibonacci spiral is the piecewise-circular curve formed by inscribing a quarter-circle of radius FnF_n in each square, with successive arcs sharing a common tangent at their junction points. As n→∞n \to \infty, this spiral converges to the logarithmic golden spiral whose growth factor per quarter turn is φ=1+52\varphi = \frac{1+\sqrt{5}}{2}.

How It Works

Start with two unit squares side by side. Attach a 2×22 \times 2 square along the longer edge of the resulting rectangle, then a 3×33 \times 3 square along the new longer edge, and so on — each new square has a side length equal to the sum of the two previous ones. Inside each square, draw a quarter-circle arc from one corner to the opposite corner. When you line these arcs up so each one starts where the last ended, the result is a smooth-looking spiral. The spiral grows by roughly a factor of φ≈1.618\varphi \approx 1.618 every quarter turn, which is why it resembles the true golden spiral.

Worked Example

Problem: Draw the first six stages of a Fibonacci spiral and find the total arc length.
Step 1: List the first six Fibonacci numbers for the square side lengths.
F1=1,  F2=1,  F3=2,  F4=3,  F5=5,  F6=8F_1 = 1,\; F_2 = 1,\; F_3 = 2,\; F_4 = 3,\; F_5 = 5,\; F_6 = 8
Step 2: Each quarter-circle arc has length equal to one-quarter of the circumference of a circle with radius FnF_n.
ℓn=2πFn4=πFn2\ell_n = \frac{2\pi F_n}{4} = \frac{\pi F_n}{2}
Step 3: Sum the six arc lengths.
L=π2(1+1+2+3+5+8)=20π2=10π≈31.4L = \frac{\pi}{2}(1+1+2+3+5+8) = \frac{20\pi}{2} = 10\pi \approx 31.4
Answer: The total arc length through six quarter turns is 10π≈31.410\pi \approx 31.4 units.

Why It Matters

The Fibonacci spiral appears in nature — sunflower seed heads, nautilus shells, and hurricane cloud bands all exhibit similar logarithmic growth. In design and architecture, the spiral guides composition and layout based on the golden ratio. Understanding it also provides a concrete entry point into studying logarithmic spirals and exponential growth in precalculus and calculus courses.

Common Mistakes

Mistake: Treating the Fibonacci spiral as identical to the golden spiral.
Correction: The Fibonacci spiral is piecewise-circular (made of quarter-circle arcs), while the true golden spiral is a smooth logarithmic spiral. The Fibonacci version only approximates the golden spiral, and the approximation improves as more squares are added.