Calculator Form
Formula Used
For an Archimedean spiral, the polar equation is:
r = a + bθ
The polar arc length formula is:
s = ∫ √(r² + (dr/dθ)²) dθ
Since dr/dθ = b, the calculator uses:
s = ∫ √((a + bθ)² + b²) dθ
The closed form result is:
s = F(θ₂) - F(θ₁)
F(θ) = [(a + bθ)√((a + bθ)² + b²) + b² asinh((a + bθ) / |b|)] / (2b)
When b = 0, the curve is a circle. The formula becomes s = |a| × |θ₂ - θ₁|.
How to Use This Calculator
- Enter the starting radius value as a.
- Enter either b per radian or pitch per full turn.
- Add the start and end angle values.
- Select the angle unit used by your input.
- Choose the input radius unit and output length unit.
- Set decimal places and Simpson intervals if needed.
- Press the calculate button to view the result.
- Use CSV or PDF export for saved records.
Example Data Table
| a | b | Start θ | End θ | Angle unit | Approximate length |
|---|---|---|---|---|---|
| 0 | 1 | 0 | 360 | Degrees | 21.256294 |
| 2 | 0.5 | 0 | 180 | Degrees | 8.894183 |
| 5 | 0 | 0 | 90 | Degrees | 7.853982 |
| 1 | 0.25 | 180 | 720 | Degrees | 28.035218 |
Arc Length of an Archimedean Spiral
An Archimedean spiral grows at a constant rate. Its polar rule is simple. Each equal angle step adds the same radial distance. That steady growth makes it useful in conversion charts, cams, coils, records, scrolls, antennas, and patterned layouts. The arc length is not just the radius times the angle. The curve also moves outward while it turns. The calculator handles both motions together.
Why This Calculator Helps
Manual work can become slow. Angles may arrive in degrees, turns, grads, or radians. Length values may use millimeters, centimeters, meters, inches, or feet. This tool converts the angle into radians first. It then applies the closed form integral. It also runs a Simpson rule check. That second value helps you compare the exact result with a numerical estimate.
Input Meaning
The value a is the radius when the angle is zero. The value b is the radial growth per radian. You may also enter pitch per full turn. In that mode, the tool divides pitch by two pi. Start and end angles define the measured part of the spiral. The order can be reversed. The reported length remains positive.
Reading Results
The main answer is the converted arc length. The result also shows start radius, end radius, angular sweep, turns, and the coefficient used by the formula. The difference between exact and Simpson values is a useful accuracy check. A small difference means the settings are consistent. More intervals usually improve the numerical check.
Good Use Cases
Use the calculator when a drawing needs a measured spiral path. Use it for classroom examples, conversion worksheets, craft templates, cutter paths, or rough engineering notes. It is also useful when a pitch is known, but the coefficient is not. The export buttons save the same inputs and outputs. CSV works well for spreadsheets. The PDF option gives a compact record. Save both for later review and audits.
Best Practice
Keep all radius values in one input unit. Choose an output unit only for the final answer. Increase decimals when working with small parts. Use radians when matching textbook formulas. Use turns when the spiral is described by complete rotations. Check the example table before entering your values.
FAQs
What is an Archimedean spiral?
It is a polar curve where radius changes at a constant rate as the angle changes. Its common equation is r = a + bθ.
What does a mean in the formula?
The value a is the starting radius when θ equals zero. It shifts the spiral inward or outward from the origin.
What does b mean in the formula?
The value b is the radial growth for each radian. A larger b makes the spiral expand faster as it turns.
Can I use degrees instead of radians?
Yes. Enter degrees and choose the degree option. The calculator converts your angle values into radians before applying the formula.
What is pitch per full turn?
Pitch is the radial distance added after one complete rotation. The calculator converts pitch into b by dividing it by 2π.
Why is Simpson check included?
It gives a numerical estimate of the same length. It helps compare the exact formula with a separate approximation method.
Can the end angle be smaller than the start angle?
Yes. The calculator measures the path length between both angles. The final arc length is always reported as positive.
What units should I use?
Use one unit for radius and growth inputs. Then select any supported output unit for the final converted arc length.