Polar Arc Length Calculator

Master polar coordinate distances with our advanced integral solver for r(θ) functions.

Show Steps

Polar Formula

L = \int_\alpha^\beta \sqrt{r^2 + (dr/d\theta)^2}\, d\theta

Equation r(theta)

Angle Start (a)

Angle End (b)

Polar Arc Length Formula

This polar arc length calculator is built for curves written as r(θ). It is especially useful for spirals, petals, and radial designs where Cartesian form is inconvenient.

L = \int_{\alpha}^{\beta} \sqrt{r^{2} + \left(\frac{dr}{d\theta}\right)^{2}}\,d\theta

Arc growth depends on both radial distance and radial change with angle.

Figure 1. Polar Arc Segment Construction

Textbook note: the integrand combines radial size r and radial rate dr/dθ.

Where Polar Arc Length Is Most Useful

Polar mode is excellent for patterns and devices naturally described by angle and radius. It avoids messy conversion to x-y equations.

Spiral paths and coil-like geometries.
Rose curves, cardioids, and antenna-lobe style equations.
Any design where angular sweep is the primary control variable.

Input And Accuracy Checklist

Use radians: keep θ in radians for derivative consistency.
Set clear bounds: choose α and β for the exact section only.
Check continuity: split the interval if the curve has breaks or singular points.
Validate with constant-radius case: for r=R, length should reduce to R(β-α).

How To Interpret The Output

The returned value is distance along the traced polar path. Increasing angular interval usually increases length, but rapid radial oscillation can increase it even faster through the derivative term.

Worked Example (Constant Radius Check)

Let r(\theta)=4 from $\theta=0$ to $\theta=\pi/3$. Then dr/d\theta = 0, and the formula simplifies naturally.

$L=\int_{0}^{\pi/3}\sqrt{4^2+0^2}\,d\theta$
$L=\int_{0}^{\pi/3}4\,d\theta=\frac{4\pi}{3}$
This matches the circle-arc identity $L=r\theta$ , which is a useful validation check.

Common Mistakes in Polar Arc Length

Degree input without conversion: keep angular math in radians unless your expression already handles conversion.
Missing derivative term: both r^2 and (dr/d\theta)^2 are required inside the root.
Negative-radius confusion: polar plotting may flip direction; confirm intended traced region.
Incorrect interval direction: check start and end angles match the physical sweep you want.

Practical Use Cases

Antenna and sensor lobe boundary length estimates in polar form.
Spiral path planning for milling, winding, and decorative manufacturing.
Analyzing petal-like biological or mechanical outlines captured as radial functions.

Related Arc-Length Tools

∿ Parametric Arc Length Σ Numerical Arc Length

Polar Tool

Polar Arc Length FAQs

What is the polar arc length formula? +

For $r(\theta)$ from $\alpha$ to $\beta$, use $L = \int_{\alpha}^{\beta} \sqrt{r^{2} + \left(\frac{dr}{d\theta}\right)^{2}}\,d\theta$.

Do I have to use radians for theta? +

Yes, radians are required for correct derivative and integration behavior in polar calculations.

Can polar arc length handle negative r values? +

Yes. The formula includes r², so sign changes in r are handled mathematically.

How do I choose theta bounds? +

Use bounds that trace exactly the portion of the curve you want, such as one petal of a rose curve.

Is polar arc length related to parametric form? +

Yes. Polar equations can be rewritten parametrically, and both approaches yield the same length.

Why is $dr/d\theta$ included in the formula? +

Arc growth depends on both radial change and angular sweep, so both terms must be included.

Can I compute spiral lengths with this mode? +

Yes. Polar mode is especially useful for spirals and radial growth curves.

How do I validate a simple polar result? +

For constant $r=R$, length should reduce to $R(\beta-\alpha)$.

What if the curve has breaks in the interval? +

Split the interval into continuous pieces, then sum each piece length.

What is a common polar input mistake? +

Using degree-style expressions while treating theta as radians.

View All Tool FAQs