Area of polar curves

Area of polar curves

In this section, we will learn how to find the area of polar curves. For polar curves, we do not really find the area under the curve, but rather the area of where the angle covers in the curve. Note that not only can we find the area of one polar equation, but we can also find the area between two polar equations. It is important to always draw the curves out so that you can locate the area you are integrating, and write the integral correctly. We will first examine a generalized formula in finding areas of polar curves. Then we will apply this to finding areas with one polar curve and then two polar curves. Make sure you know your trigonometric identities very well before tackling these questions.


To find the area of a single polar equation, we use the following formula:
where α\alpha is the starting angle and β\beta is the ending angle.

To find the area that is enclosed by two polar equations like in the picture below, we use the formula:
A=αβ12(r2  outerr2  inner)dθA=\int_{\alpha}^{\beta}\frac{1}{2}(r^{2}\;_{outer}-r^{2}\;_{inner})d\theta
where routerr_{outer} is the outer part of the first polar equation, and rinnerr_{inner} is the inner part of the second polar equation.
area of polar curves
  • Introduction
    Area of Polar Curve Overview

  • 1.
    Area Enclosed with One Polar Equation
    Find the area of the inner loop of r=1+2sinθr=1+2\sin\theta.

  • 2.
    Find the area of the inner loop of r=12cosθr=1-2\cos\theta.

  • 3.
    Area Enclosed by two Polar Equations
    Find the Area enclosed outside of r=4r=4 and inside r=9r=9.

  • 4.
    Find the Area enclosed outside r=1r=1 and inside r=1+sinθr=1+\sin\theta.