How to Graph Polar Equations

By Education Editor

How to Graph Polar Equations. Polar equations are math functions given in the form of R= f (θ). To express these functions you use the polar coordinate system. The graph of a polar function R is a curve that consists of points in the form of ( R, θ). Due to the circular aspect of this system, it's easier to graph polar equations using this method.

Understand Polar Equations

Understand that in the polar coordinate system you denote a point by (R, θ) where R is the polar distance and θ is the polar angle in degrees.

Use radian or degrees to measure θ. To convert radians to degrees, multiply the value by 180/π. For example, π/2 X 180/π = 90 degrees.

Know that there are many curve shapes given by polar equations. Some of these are circles, limacons, cardioids and rose-shaped curves. Limacon curves are in the form R= A ± B sin(θ) and R= A ± B cos(θ) where A and B are constants. Cardioid (heart-shaped) curves are special curves in the limacon family. Rose petalled curves have polar equations in the form of R= A sin(nθ) or R= A cos(nθ). When n is an odd number, the curve has n petals but when n is even the curve has 2n petals.

Simplify the Graphing of Polar Equations

Look for symmetry when graphing these functions. As an example use the polar equation R=4 sin(θ).You only need to find values for θ between π (Pi) because after π the values repeat since the sine function is symmetrical.

Choose the values of θ that makes R maximum, minimum or zero in the equation. In the example given above R= 4 sin (θ), when θ equals 0 the value for R is 0. So (R, θ) is (0, 0). This is a point of intercept.

Find other intercept points in a similar manner.

Graph Polar Equations

Consider R= 4 sin(θ) as an example to learn how to graph polar coordinates.

Evaluate the equation for values of (θ) between the interval of 0 and π. Let (θ) equal 0, π /6 , π /4, π /3, π /2, 2π /3, 3π /4, 5π /6 and π. Calculate values for R by substituting these values into the equation.

Use a graphing calculator to determine the values for R. As an example, let (θ) = π /6. Enter into the calculator 4 sin(π /6). The value for R is 2 and the point (R, θ) is (2, π /6). Find R for all the (θ) values in Step 2.

Plot the resulting (R, θ ) points from Step 3 which are (0,0), (2, π /6), (2.8, π /4), (3.46,π /3), (4,π /2), (3.46, 2π /3), (2.8, 3π /4), (2, 5π /6), (0, π) on graph paper and connect these points. The graph is a circle with a radius of 2 and center at (0, 2). For better precision in graphing, use polar graph paper.

Graph the equations for limacons, cardioids or any other curve given by a polar equation by following the procedure outlined above.


Note that the topic on graphing polar equation is extensive and there are many other curve shapes then the ones mentioned here. Please look at the resources for more information on graphing these. A quicker method to graph polar equations is to use a hand-held graphing calculator or an online graphing calculator. Graphing polar functions produces intricate curves so it is best to graph them by plotting points.