Spherical Coordinates Calculator

About Spherical Coordinates Calculator

Spherical coordinates (r, θ, φ) specify a point by distance from origin, polar angle from z-axis, and azimuthal angle in the xy-plane.

How to use this calculator

1. Enter the Cartesian values \(x\), \(y\), and \(z\) if you want to convert from Cartesian to spherical.
2. Or enter the spherical values, usually \(r\), \(\theta\), and \(\phi\), if you want to convert back to Cartesian.
3. Check the calculator output carefully, especially the angle units if your class uses degrees or radians.
4. Use the result to rewrite points, graph positions, or simplify geometry and physics problems.

The formula explained

The formula \(r = \sqrt{x^2+y^2+z^2}\) finds the distance from the origin to a point \((x,y,z)\). In spherical coordinates, that distance is the radial coordinate \(r\).

• x = the Cartesian coordinate along the \(x\)-axis
• y = the Cartesian coordinate along the \(y\)-axis
• z = the Cartesian coordinate along the \(z\)-axis
• r = the distance from the origin to the point
• \(\theta\) = an angle used in spherical coordinates, often measured in the \(xy\)-plane
• \(\phi\) = an angle used in spherical coordinates, often measured from the positive \(z\)-axis

Step by step method

1. Start with the Cartesian point \((x,y,z)\).
2. Find the radius using \(r = \sqrt{x^2+y^2+z^2}\).
3. Use the point's horizontal projection and angles to determine the remaining spherical coordinates.
4. Check that the angles match the calculator's convention before using the result.

Worked example

Problem. Convert the Cartesian point \((2,2,1)\) to spherical coordinates.

1. Compute the radius, \(r = \sqrt{2^2+2^2+1^2} = \sqrt{9} = 3\).
2. Find the horizontal distance, \(\sqrt{x^2+y^2} = \sqrt{2^2+2^2} = \sqrt{8} = 2\sqrt{2}\).
3. Use the angles, \(\theta = \tan^{-1}(2/2) = \tan^{-1}(1) = \frac{\pi}{4}\), and \(\phi = \cos^{-1}(1/3)\).

Tips and common mistakes

• Make sure you know which angle convention your course uses, because spherical coordinates can be defined in more than one way.
• Do not forget to square all three Cartesian coordinates when finding \(r\), then take the square root at the end.

Frequently asked questions