Analyze conical spiral geometry with practical physics inputs. Calculate path length, pitch, cone angle, spacing. Review formulas, example data, exports, and clear usage steps.
| Starting Radius | Ending Radius | Height | Turns | Arc Length | Pitch | Half Angle |
|---|---|---|---|---|---|---|
| 6 cm | 2 cm | 15 cm | 4 | 101.835421 cm | 3.750000 cm | 14.931417 degrees |
Radius function: r(t) = r1 + ((r2 - r1) / (2πn)) × t
Height function: z(t) = (h / (2πn)) × t
Position model: x(t) = r(t) cos(t), y(t) = r(t) sin(t), z(t) = z(t)
Pitch per turn: p = h / n
Radial change per turn: Δr = (r2 - r1) / n
Slant height: s = √(h² + (r2 - r1)²)
Cone half angle: α = tan⁻¹(|r2 - r1| / h)
Arc length: L = ∫ √(r(t)² + (dr/dt)² + (dz/dt)²) dt
This file estimates arc length numerically with Simpson's rule.
A conical spiral is a three dimensional curve wrapped around a cone. The radius changes while the path rotates upward. This makes it useful in physics, motion design, and geometric modeling.
A normal helix keeps a constant radius. A conical spiral does not. Its radius expands or contracts with height. That difference changes arc length, spacing, and slope. Engineers and physics students study this path when analyzing tapered coils, scanning paths, and rotating motion on conical surfaces.
This calculator estimates the most useful geometric values. It finds spiral length, pitch per turn, radial change per turn, average radius, slant height, and cone half angle. These outputs help when planning motion, material length, or surface travel. They also make comparison between compact and wide spirals easier.
The path can be described with parametric equations. Radius changes linearly from the starting radius to the ending radius. Height also changes linearly through the selected number of turns. Rotation is measured with angle. From these parts, the calculator builds the full three dimensional position and then estimates total path length.
Students can use this tool for geometry assignments and physics practice. Designers can test how shape changes affect path distance. Fabrication teams can estimate wire or track length on tapered forms. Researchers can compare different cone heights and turn counts without manual repetition.
Small input changes can shift the final length noticeably. More turns usually increase the path length. Greater height increases pitch. A larger radius difference changes the cone angle and slant geometry. For reliable results, keep units consistent across every field. The unit label is optional, but it helps make reports clearer.
The result section appears immediately after calculation. Export options help save the current report as CSV or PDF. The example table gives a quick reference point. Formula notes and usage steps support learning, checking, and repeat work with less effort.
Because this method estimates arc length numerically, dense sampling improves stability. That provides dependable values for classwork, prototypes, reports, and repeated verification across practical daily cases.
A conical spiral is a curve that wraps around a cone while its radius changes with height. It differs from a standard helix because the radius is not constant.
It returns arc length, pitch per turn, radial change per turn, average radius, slant height, cone half angle, circumferences, and total rotation.
The curve changes radius continuously. Numerical integration gives a stable practical estimate for the full three dimensional path length.
Use any unit you want, but keep every length input in the same unit. The optional unit label only helps display and export the report clearly.
Pitch is the vertical rise for one full turn. It equals total height divided by the number of turns.
Yes. Enter a smaller ending radius than the starting radius. The calculator handles expanding and contracting conical spirals.
It shows the taper of the cone relative to its axis. A larger angle means the cone opens or closes more quickly over the given height.
Increase segments when you want a finer arc length estimate, especially for long spirals, many turns, or cases needing repeatable accuracy.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.