How to graph circles in polar coordinates

A polar circle can look very different depending on where its center is. A constant radius puts the center at the pole, while a sine or cosine radius can move it away from the pole. Let’s connect those forms with the circles they describe.

Recognize a constant-radius circle

For r=a with nonzero a, the distance from the pole is always |a|. Both r=2 and r=-2 trace the circle of radius 2 centered at the origin. The equation r=0 gives the pole itself.

Find the center of a cosine circle

For r=-7cosθ, multiply by r and use rcosθ=x. Completing the square gives

x2+y2=-7x(x+72)2+y2=494

The center is (-72,0) and the radius is 72. At θ=0, the negative radius -7 plots the point (-7,0). At θ=π2, the radius is zero, so the curve reaches the pole.

r = −7 cos θA Cartesian coordinate grid shows the polar curve r = −7 cos θ. The numbered grid values are outside the plotted region.r = −7 cos θ−8−6−4−202−4−2024xy
The circle is centered at (−3.5, 0), has radius 3.5, and passes through the pole.

Use the same conversion for a sine circle

For r=-10sinθ, multiplying by r gives x2+y2=-10y. Completing the square gives x2+(y+5)2=25. The center is (0,-5) and the radius is 5.

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