Polar Coordinates: Plotting Points on the (r, θ) Grid

Polar coordinates describe a point's location using a distance from the origin ($r$) and an angle from the positive x-axis ($\theta$), instead of the Cartesian $(x, y)$.

Conversion formulas

Polar → Cartesian: $x = r\cos\theta$, $y = r\sin\theta$

Cartesian → Polar: $r = \sqrt{x^{2} + y^{2}}$, $\theta = \arctan(y/x)$ (adjusted for quadrant)

Plotting the three points

$(3, \pi/4)$: $r = 3$ at $45°$. $x = 3\cos 45° = 2.12$, $y = 3\sin 45° = 2.12$. Upper-right quadrant.

$(5, 2\pi/3)$: $r = 5$ at $120°$. $x = 5\cos 120° = -2.5$, $y = 5\sin 120° = 4.33$. Upper-left quadrant.

$(-2, \pi/6)$: Negative $r$ means go opposite direction. $r = -2$ at $30°$ means go 2 units at $30° + 180° = 210°$. $x = -2\cos 30° = -1.73$, $y = -2\sin 30° = -1.0$. Lower-left quadrant.

When to use polar coordinates

Polar coordinates shine when the geometry has circular symmetry — orbits, waves, antennas, radar, spirals. The equation of a circle of radius 5 centered at the origin is simply $r = 5$ in polar (vs. $x^{2} + y^{2} = 25$ in Cartesian).

Try it in the visualization

Click to place points on the polar grid. The $(r, \theta)$ and $(x, y)$ coordinates are shown simultaneously. Toggle "show conversion" to see the conversion formulas applied. Drag the angle slider to sweep a ray and understand how $\theta$ works.