How this instrument works
An Archimedean spiral is the curve traced by a point moving outward from a center at a constant rate while turning at a constant angular speed, described by the simple relation r = a·θ: the radius r grows in direct proportion to the angle θ swept, scaled by a growth rate a. Sweep twice the angle and the spiral has moved twice as far outward, no more and no less.
That constant growth rate is what sets it apart from a logarithmic spiral, whose radius instead grows exponentially with angle, spacing its turns ever wider apart the further out it goes. An Archimedean spiral keeps its successive turns an even, fixed distance apart, the same spacing near the center as far from it, which is exactly the pattern found in a rolled-up garden hose or ribbon, a coiled rope, and the grooves cut into a vinyl record.
Because the curve bends as it grows, its arc length, the distance covered tracing it with a pencil, is always longer than a straight line drawn between its start and end points. Working that out takes more than plain geometry: the closed-form expression involves a square root and a natural logarithm, both handled here, so entering a growth rate and a swept angle returns the exact traced distance rather than a rough estimate.
Some spiral staircases follow a related winding logic, gaining a roughly constant amount of horizontal reach with every step around the center pole. Picture the Archimedean spiral as the idealized, flat version of that same steady, constant-rate winding, stripped down to a single equation.
- Enter the spiral's growth rate in the Spiral growth rate (a) field — this sets how fast the radius grows per radian swept.
- Enter the total angle swept, in radians, in the Angle swept (θ) field.
- Read Arc length for the exact distance traced from the center out to that angle.
- Try θ = 2π, about 6.283, for one full turn, or π, about 3.1416, for half a turn.
- Set Angle swept (θ) to 0 to confirm the spiral has traced no distance yet at its own starting point.
Worked example — one full turn
With growth rate a=1 and a full turn swept, θ=2π ≈ 6.283185307179586 radians, the formula gives L = (1/2)(θ√(1+θ²) + ln(θ+√(1+θ²))) ≈ 21.256294148209097 units, noticeably more than the roughly 6.283 units of straight-line radius reached at that angle, since the path curves the entire way out rather than running straight.
Halve the turn and double the growth rate, a=2, θ=π ≈ 3.141592653589793, and the same formula returns L ≈ 12.219838667918586 units. At the very start of any spiral, θ=0, no angle has been swept and no distance has been traced, so L=0 exactly, whatever value a takes.
Questions
What makes a spiral 'Archimedean' rather than logarithmic?
A constant growth rate — the radius r = a·θ increases by the same fixed amount for every additional radian swept, so consecutive turns stay an even distance apart. A logarithmic spiral instead grows exponentially with angle, so its turns spread further and further apart the more it winds outward.
Where does this kind of spiral show up in everyday life?
In anything wound at a steady rate: the grooves cut into a vinyl record, a garden hose or ribbon rolled into neat coils, and some spiral staircases, which gain roughly the same amount of outward reach with each turn around the central pole. All of them share the same constant-spacing pattern as r = a·θ.
Why is the arc length formula more complicated than the radius formula?
Because arc length measures distance along a curving path, not a straight one, and that requires accounting for the spiral's changing direction as it winds. The result, involving a square root and a natural logarithm, is exact rather than approximate; it is not a shortcut, just the closed form the underlying calculation happens to simplify to.
Is the arc length ever shorter than the straight-line distance to the same point?
No. A curving path between two points is always at least as long as the straight line connecting them, and typically longer, since the spiral bends continuously rather than heading directly outward. Sweeping a full turn at a=1 traces about 21.26 units of curve while the endpoint sits only about 6.28 units from the center.
What happens to the arc length if the growth rate a doubles?
Arc length scales directly with a for any fixed angle, since a appears as a simple multiplying factor out front of the formula. Doubling the growth rate while keeping the same swept angle exactly doubles the traced length, the same proportional relationship the radius itself follows.
What is the arc length at the very start of the spiral?
Zero. At θ=0 no angle has been swept yet, so the point sits at the spiral's own center and no distance has accumulated at all, regardless of the growth rate entered. Length only begins building once θ moves above zero.