Archimedean Spiral — Definition, Formula & Examples

An Archimedean spiral is a curve that winds outward from a central point at a constant rate, so each successive loop is the same distance from the previous one.

In polar coordinates, an Archimedean spiral is the locus of points satisfying $r = a + b\theta$ , where $a$ and $b$ are real constants and $\theta \geq 0$ . The parameter $b$ controls the uniform spacing between consecutive turns, which equals $2\pi |b|$ .

Key Formula

r = a + b\theta

Where:

$r$ = Distance from the origin (pole) to a point on the spiral
$\theta$ = Angle measured from the polar axis, in radians
$a$ = Initial radius when θ = 0; shifts the spiral away from the origin
$b$ = Rate at which the spiral expands per radian; spacing between turns is 2π|b|

How It Works

To graph an Archimedean spiral, you plot points in polar coordinates by choosing values of

\theta

and computing

r = a + b\theta

. As

\theta

increases,

r

grows linearly, so the spiral expands outward at a steady pace. Unlike a logarithmic spiral, the gaps between successive arms remain constant. Setting

a = 0

places the starting point at the origin; a nonzero

a

shifts the starting radius outward.

Worked Example

Problem: Find the polar coordinates of three points on the Archimedean spiral r = 2θ, and determine the distance between consecutive turns.

Evaluate at θ = 0: Substitute θ = 0 into the equation.

r = 2(0) = 0

Evaluate at θ = π: Substitute θ = π.

r = 2\pi \approx 6.28

Evaluate at θ = 2π: Substitute θ = 2π. This completes one full revolution.

r = 2(2\pi) = 4\pi \approx 12.57

Find the spacing: The gap between consecutive turns equals 2π|b|.

\text{spacing} = 2\pi(2) = 4\pi \approx 12.57

Answer: The three points are (0, 0), (2π, π), and (4π, 2π). Each full revolution adds 4π ≈ 12.57 units to the radius.

Why It Matters

Archimedean spirals appear in vinyl record grooves, watch springs, and scroll compressors — anywhere uniform spacing during winding matters. In calculus, they provide excellent practice for computing arc length and area in polar coordinates.

Common Mistakes

Mistake: Confusing an Archimedean spiral with a logarithmic spiral.

Correction: In an Archimedean spiral, r grows linearly with θ, so turns are equally spaced. In a logarithmic spiral, r grows exponentially, so the spacing between turns increases outward.