In the previous sections, area and arc length based on the rectangular coordinates or in parametric form has been presented. The area and arc length in polar coordinates will be introduced in this section.

Area Calculation with
Polor Coordinates

Area of the Left Loop of the
Four-Leaved Rose Curve

When a curve is expressed in polar coordinates, r = f(θ), the area bounded by θ = a and θ = b is given by

As an example, the area for one leaf of the four-leaved rose can be represented by r = cos(2θ) and bounded by θ = 3π/4 and θ = 5π/4.

Arc Length

When a curve is given in terms of polar coordinates r = f(θ) for a ≤ θ ≤ b, the equation can be rewritten in the parametric form as follows:

     x = rcosθ = f(θ)cosθ   and   y = rsinθ = f(θ)sinθ

where θ is the parameter.

According to the definition of an arc length introduced previously, the arc length of a polar curve is determined as,

Take the curve of a cardioid as an example. The polar curve is represented by r = 1 + cosθ. The arc length is,