How to use the coordinate system converter
- Choose the system you have: Cartesian 2D, polar, Cartesian 3D, cylindrical or spherical.
- Choose degrees or radians and enter the values. The fields change with the system.
- Read the headline for the main conversion and the table for the point in every system.
- For spherical coordinates, check which angle your source lists first, because the table gives both orders.
Formula
Polar to Cartesian: x = r cos(angle), y = r sin(angle)
Cartesian to polar: r = square root of (x^2 + y^2), angle = atan2(y, x)
Spherical to Cartesian: x = r sin(polar) cos(azimuth), y = r sin(polar) sin(azimuth), z = r cos(polar)
Cartesian to spherical: r = square root of (x^2 + y^2 + z^2), polar = arccos(z / r), azimuth = atan2(y, x)
Cylindrical coordinates are polar coordinates in the xy plane plus the height z. The polar angle of a spherical point is measured down from the positive z axis, from 0 to 180 degrees.
Three conversions
The polar point (2, 30 degrees) is x = 2 cos 30 = 1.7320508 and y = 2 sin 30 = 1, so (1.7320508, 1). The point (3, 4) has r = 5 and an angle of 53.130102 degrees. The point (-1, -1) has r = 1.4142136 and an angle of 225 degrees, because angles run from 0 up to a full turn.
The 3D point (1, 1, 1) has a radius of 1.7320508, a polar angle of 54.73561 degrees from the z axis and an azimuth of 45 degrees. The spherical point (5, 60 degrees, 45 degrees) is (3.0618622, 3.0618622, 2.5) in Cartesian form.
Which convention is yours
Physics and engineering texts usually write spherical coordinates as radius, polar angle from the z axis, then azimuth. Many math texts write radius, azimuth, then polar angle, and some swap the names theta and phi. Because of this the table lists both orders. Always check the definition in your source before comparing numbers.
For vectors and angles between them, see the vector calculator. The trigonometry calculator handles sine, cosine and tangent, and the triangle calculator solves triangles from their sides and angles.
Edge cases
At the origin the angles are not defined, so the page reports them as 0. Angles come out from 0 up to a full turn, so a point in the lower half of the plane has an angle above 180 degrees instead of a negative one. Tiny rounding residues, such as the 6e-17 you get from cos(90 degrees), are cleaned to zero.
Assumptions and limits
- The coordinate axes are right handed, with z pointing up from the xy plane.
- Radii must be zero or more. The polar angle of a spherical point must be from 0 to 180 degrees.
- Angles are reported from 0 up to a full turn, in the unit you chose.
- Values are shown with up to 8 significant digits.
Frequently asked questions
How do I convert Cartesian coordinates to polar?
The radius is the square root of x squared plus y squared, and the angle is atan2(y, x). The point (3, 4) has a radius of 5 and an angle of about 53.13 degrees.
What is the difference between cylindrical and spherical coordinates?
Cylindrical uses the distance from the z axis, an azimuth angle and the height z. Spherical uses the distance from the origin, a polar angle from the z axis and the azimuth.
Why do some books swap theta and phi?
Physics usually uses theta for the polar angle and phi for the azimuth, and many math books do the opposite. The page lists both orders so you can match your source.
Can I use radians?
Yes. Switch the angle unit to radians, and both the inputs and the results use radians.