What is the arc length of the Archimedean Spiral?

Arc length (from 0 to θ, a = 0): L = (b/2)[θ√(1+θ^2) + ln(θ + √(1+θ^2))]. a is the starting radius (a = 0 in the graph) and b sets the gap between turns.

What is the area of the Archimedean Spiral?

Area (first turn, a = 0): A = (4/3)π^3 b^2. a is the starting radius (a = 0 in the graph) and b sets the gap between turns.

Spirals

Archimedean Spiral

The constant-pitch spiral r = a + bθ, whose successive turns stay exactly 2πb apart.

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b (spacing / 2π)0.16turns4

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Formulas and key values

Polar equation: $r = a + b\theta$
Gap between turnsIn the graph ≈ 1.005: $2\pi b$
Arc length · from 0 to θ, a = 0In the graph ≈ 50.89: $L = \frac{b}{2}\left[\theta\sqrt{1+\theta^2} + \ln\!\left(\theta + \sqrt{1+\theta^2}\right)\right]$
Area · first turn, a = 0In the graph ≈ 1.058: $A = \frac{4}{3}\pi^3 b^2$

a is the starting radius (a = 0 in the graph) and b sets the gap between turns.

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Make a video of the Archimedean Spiral

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01 · Definition

What is the Archimedean Spiral?

The Archimedean spiral r = a + bθ grows outward at a constant rate: every full turn adds exactly 2πb to the radius, so the gap between coils never changes. Archimedes analyzed it in his treatise On Spirals around 225 BC, using it to square the circle and trisect angles — two feats impossible with compass and straightedge alone.

Because the spacing is uniform, this is the spiral of wound-up physical things: rolled paper, vinyl record grooves, watch springs, and the scroll compressor inside many air conditioners.

02 · Properties

Key properties

Successive turns are separated by a constant distance 2πb.
A ray from the origin meets consecutive coils at equal spacing — the defining property.
Arc length from 0 to θ: (b/2)[θ√(1+θ²) + ln(θ + √(1+θ²))] for r = bθ.
The area swept by the first turn of r = bθ is (4/3)π³b² (Archimedes’ own result: one third of the enclosing circle).
Polar slope angle grows with θ — the spiral becomes ever more circle-like far from the center.

03 · Questions