Polar curve constant r

By Martin McBride, 2020-09-18
Tags: standard curves
Categories: polar coordinates

Here is an example of a polar function of constant $r$.

$$r = 3$$

This function describes a curve where $r$ is constant for any angle $\theta$, in this case 3. The curve represents the locus all the points that are a distance of exactly 3 from the origin. This curve, of course, is a circle of radius 3:

This animation shows how the curve is plotted as $\theta$ moves from 0 to $2\pi$:

Varying the radius

In general the function:

$$r = n$$

where $n$ is a non-negative constant, describes a circle of radius $n$. Here are the circles created for $n = 3$ (red), $n = 2$ (green), and $n = 1$ (blue):

Cartesian equation

We can convert the polar equation into a Cartesian equation using the identity:

$$r^2 = x^2 + y^2$$

Substituting for $r$ in the equation above gives:

$$x^2 + y^2 = n^2$$

which is the Cartesian equation for a circle of radius $n$, centred on the origin.

See also

Join the GraphicMaths Newletter

Sign up using this form to receive an email when new content is added:

Popular tags

angle area cartesian equation chord circle combinations complex polygon cosh cosine cosine rule cube decagon diagonal directrix dodecagon ellipse equilateral triangle exponential exterior angle focus hendecagon heptagon hexagon horizontal hyperbola hyperbolic function interior angle inverse hyperbolic function irregular polygon isosceles trapezium isosceles triangle kite locus major axis minor axis nonagon normal octagon parabola parallelogram parametric equation pentagon perimeter permutations power quadrilateral radius rectangle regular polygon rhombus root sine rule sinh sloping lines solving equations solving triangles square standard curves star polygon straight line graphs symmetry tangent tanh transformations trapezium triangle vertical