Spiral — Definition, Formula & Graph

Spiral

A curve on a plane that turns endlessly outward or inward (or both). Spirals usually have polar equations. A few of the many types of spirals are pictured below.

Four spiral types: Archimedean (r=aθ), Hyperbolic (rθ=a), Logarithmic (r=e^θ), and Parabolic ((r-a)²=4akθ).

See also

Golden spiral, polar curves

Key Formula

r = a + b\theta

Where:

$r$ = Distance from the origin (pole) to a point on the spiral
$\theta$ = Angle in radians measured from the positive x-axis
$a$ = Starting radius — the distance from the origin when θ = 0
$b$ = Rate of expansion — controls how quickly the spiral moves outward per radian

Worked Example

Problem: Plot key points of the Archimedean spiral r = 1 + 2θ for θ = 0, π/2, π, 3π/2, and 2π.

Step 1: Identify the parameters: a = 1 (starting radius) and b = 2 (expansion rate).

r = 1 + 2\theta

Step 2: Compute r at θ = 0.

r = 1 + 2(0) = 1

Step 3: Compute r at θ = π/2 ≈ 1.571.

r = 1 + 2\!\left(\frac{\pi}{2}\right) = 1 + \pi \approx 4.14

Step 4: Compute r at θ = π and θ = 3π/2.

r(\pi) = 1 + 2\pi \approx 7.28, \qquad r\!\left(\tfrac{3\pi}{2}\right) = 1 + 3\pi \approx 10.42

Step 5: Compute r at θ = 2π (one full revolution).

r = 1 + 2(2\pi) = 1 + 4\pi \approx 13.57

Answer: After one full revolution, the spiral has expanded from r = 1 to r ≈ 13.57. Each successive loop is separated by a constant radial distance of 2π · b = 4π ≈ 12.57 units.

Another Example

This example uses a logarithmic spiral (r = ae^(bθ)) instead of the Archimedean spiral (r = a + bθ), showing how exponential growth produces unequally spaced loops.

Problem: Find polar coordinates of points on the logarithmic (exponential) spiral r = 2e^(0.1θ) at θ = 0, π, 2π, and 4π.

Step 1: The logarithmic spiral has the form r = ae^(bθ). Here a = 2 and b = 0.1.

r = 2e^{0.1\theta}

Step 2: At θ = 0:

r = 2e^{0} = 2

Step 3: At θ = π ≈ 3.14:

r = 2e^{0.1\pi} = 2e^{0.3142} \approx 2(1.369) \approx 2.74

Step 4: At θ = 2π and θ = 4π:

r(2\pi) = 2e^{0.6283} \approx 3.75, \qquad r(4\pi) = 2e^{1.2566} \approx 7.03

Step 5: Notice that the ratio of r values at successive full revolutions is constant: r(4π)/r(2π) ≈ 7.03/3.75 ≈ 1.875 and r(2π)/r(0) ≈ 3.75/2 = 1.875. This constant ratio is the hallmark of a logarithmic spiral.

\frac{r(\theta + 2\pi)}{r(\theta)} = e^{0.1 \cdot 2\pi} = e^{0.2\pi} \approx 1.875

Answer: The points are (2, 0), (2.74, π), (3.75, 2π), and (7.03, 4π). Unlike the Archimedean spiral, the gap between loops grows because the expansion is multiplicative, not additive.

Frequently Asked Questions

What is the difference between an Archimedean spiral and a logarithmic spiral?

An Archimedean spiral has the equation r = a + bθ, so successive loops are separated by a constant distance (2πb). A logarithmic spiral has the equation r = ae^(bθ), so each loop is a constant multiple wider than the previous one. Archimedean spirals expand at a steady, linear rate; logarithmic spirals expand exponentially, which is why they appear in nature (e.g., nautilus shells).

Is a spiral the same as a helix?

No. A spiral is a flat (two-dimensional) curve that winds around a point on a plane. A helix is a three-dimensional curve that winds around an axis while also moving along that axis — think of a corkscrew or a coiled spring. Every helix rises in space; a spiral stays in one plane.

Why are polar coordinates used for spirals instead of Cartesian?

Spirals naturally wind around a central point, so describing them in terms of a radius r and an angle θ is far simpler. In Cartesian form, the same curve would require complicated expressions relating x and y. Polar equations like r = a + bθ capture the winding behavior directly.

Archimedean Spiral vs. Logarithmic Spiral

	Archimedean Spiral	Logarithmic Spiral
Polar equation	r = a + bθ	r = ae^(bθ)
Growth type	Linear — constant distance between loops	Exponential — constant ratio between loops
Loop spacing	Equal (spacing = 2πb)	Increases with each revolution
Self-similar?	No	Yes — looks the same at every scale
Common example	Grooves on a vinyl record	Nautilus shell, hurricane bands

Why It Matters

Spirals appear throughout the math curriculum: in polar graphing, parametric equations, and the study of sequences like the Fibonacci sequence (which is closely tied to the golden spiral). Beyond the classroom, spirals model natural phenomena — galaxy arms, cyclone patterns, and biological growth — making them one of the most practical curves you will encounter. Mastering their polar equations is essential for success in precalculus and calculus courses that cover polar coordinates.

Common Mistakes

Mistake: Confusing a spiral with a circle. Students sometimes set θ to a single value and think the curve closes on itself.

Correction: A circle has a constant r for all θ. In a spiral, r changes as θ increases, so the curve never closes — it keeps winding outward (or inward).

Mistake: Forgetting that θ must be in radians when using the standard spiral formulas.

Correction: The formulas r = a + bθ and r = ae^(bθ) assume θ is in radians. Using degrees will give incorrect r values and distort the shape of the curve.

Related Terms

Polar Equation — The coordinate system used to express spiral formulas
Polar Curves — Broader family of curves that includes spirals
Golden Spiral — A special logarithmic spiral based on the golden ratio
Curve — General term for the type of geometric object a spiral is
Plane — The flat surface on which a spiral lies
Parametric Equations — Alternative way to express spiral coordinates as x(t), y(t)
Fibonacci Sequence — Number sequence closely linked to the golden spiral