Spiral Length Calculator

An Archimedean spiral is a polar curve whose radius changes at a constant rate as its angle changes. Its equation is r = a + bθ, where a is the radius at angle zero, b is the radial growth for each radian, and θ is the angle in radians. Equal angular steps therefore produce equal changes in radius, giving neighboring turns a constant separation. This familiar spiral appears in rolled materials, grooves, coils, antennas, and mathematical models. A logarithmic spiral instead uses r = a times e raised to bθ. Its radius changes by a constant proportion over equal angular steps, so turn spacing expands geometrically rather than arithmetically. Logarithmic spirals occur in growth models, antenna design, and natural forms. This calculator supports both families and applies the appropriate exact antiderivative for the selected equation. Finding spiral length requires more than multiplying an average radius by an angle because the curve moves both around the origin and radially at the same time. For a polar curve, the arc-length integrand is the square root of r squared plus the square of the derivative of r with respect to θ. For an Archimedean spiral the derivative is simply b. For a logarithmic spiral the derivative is b times the current radius, which leads to a compact exponential length formula. The implementation evaluates closed-form expressions rather than splitting the curve into many short straight segments. When b is zero, either spiral has constant radius and becomes a circular arc; the calculator uses the simpler radius times angular span formula. Angles entered in degrees are converted to radians before any formula is applied. The initial radius and growth value must be interpreted according to the selected model. Archimedean growth carries length per radian, while the logarithmic exponential factor is dimensionless. The resulting arc length uses the initial radius unit. A negative growth describes a spiral winding inward over increasing angle, and length remains nonnegative. Real manufactured coils have thickness, deformation, and tolerances not captured by an ideal centerline model. For planning material length, use the centerline radius for a strip or wire and include a practical allowance. The six-decimal display provides a clear numerical result while the internal computation retains standard double precision.