How to Graph a Circle Using Parametric Equations

Try to graph x² + y² = 9 (a circle of radius 3) directly in Function mode and you’ll hit a wall: solving for y gives you y = ±√(9 - x²), two separate functions, and neither one draws the whole circle on its own. This is a sign you’re fighting the wrong mode, not a sign the circle is hard to graph.

Why parametric mode fixes this

Function mode requires exactly one y for every x — that’s what makes something “a function” in the first place. A circle fails this test twice over (most x values hit the circle at two different y values), which is precisely why the direct approach needs two separate equations.

Parametric mode sidesteps the whole problem by describing the curve with a third variable, usually called T, that sweeps through both x and y together:

X1T = 3cos(T)
Y1T = 3sin(T)

As T runs from 0 to 2π, the point (3cos(T), 3sin(T)) traces the full circle exactly once, radius 3, centered at the origin. There’s no case-splitting, no ±, and no missing half of the curve.

The general pattern

Any circle of radius r centered at (h, k) is:

X1T = h + r·cos(T)
Y1T = k + r·sin(T)

This same trick — introducing a parameter that both coordinates depend on — is also how you graph ellipses, spirals, and any curve that would otherwise need multiple functions or fail the vertical line test.

Try it yourself

Open the TI-84 Calculator, switch to Graph mode, and select Parametric from the mode tabs above the plot. Enter 3cos(T) and 3sin(T) into the first X1T/Y1T slot — the default T range of 0 to 2π is already set up for exactly this case. Drag to pan and scroll to zoom once it’s drawn.

Or skip the typing — the examples gallery has this exact circle one tap away, along with a few other curves worth trying.