Arc Length of Polar Curves
Main Concept

For polar curves of the form , the arc length of a curve on the interval can be calculated using an integral.

Calculating Arc Length


The x and ycoordinates of any Cartesian point can be written as the following parametric equations:
.
Using the chain rule, note that and .
For these parametric equations, the arc length of a curve on the interval is given by:
Remembering that , this expression can now be simplified to:
Therefore the arc length for a polar curve on the interval is .




Choose a polar function from the list below to plot its graph. Enter the endpoints of an interval, then use the slider or button to calculate and visualize the arc length on the given interval. When choosing the endpoints, remember to enter π as "Pi".


Interval θ =
The arc length of the curve on this interval is








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