Cartesian and Polar

Convert points and equations between Cartesian and polar coordinates with a solution. Circles, lines and other curves.

What to find:

polar coordinates of a point from x and y

Cartesian coordinates of a point from r and φ

polar equation from an equation in x and y

Cartesian equation from an equation in r and φ

Write powers as x^2. Examples: y = 2x + 1, x^2 + y^2 = 4x, r = 5, r = 4cos(φ), r = 3/cos(φ), φ = π/4. You can write phi instead of φ.

Test yourself

This converter converts points and equations between Cartesian coordinates (x; y) and polar coordinates (r; φ).

Conversion formulas

x = r cos φ, y = r sin φ, r² = x² + y², tan φ = y / x

Equations of lines and circles are simplified to r = … or tan φ = …. For other equations the converter substitutes the formulas and shows the result without simplifying it. From polar to Cartesian, it handles r = a, r = a cos φ + b sin φ, r = a / (b cos φ + c sin φ) and φ = a.

Example. The circle x² + y² = 25 in polar coordinates is r = 5.