Coordinate Converter: Rectangular, Polar, Cylindrical and Spherical Coordinates
Convert a point between coordinate systems.
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Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.
How this calculator works
Choose the conversion you want.
Enter the coordinates you know.
The calculator gives the converted coordinates, with the angle in the right quadrant.
Formulas used
- r = \sqrt{x^2 + y^2},\ \theta = \operatorname{atan2}(y, x)
- x = r\cos\theta,\ y = r\sin\theta
- \rho = \sqrt{x^2 + y^2 + z^2},\ \varphi = \arccos\frac{z}{\rho}
Worked example
With the default numbers in the calculator above:
- Radius r: 5
- Angle theta in degrees: 53.1301 degrees
- Angle theta in radians: 0.927295 rad
Change the numbers in the calculator to see the full working, every step, for your own case.
A second worked example
A different case, with these numbers: x -3; y -4.
- Radius r: 5
- Angle theta in degrees: 233.13 degrees
- Angle theta in radians: 4.06889 rad
Common questions
How do I convert rectangular to polar coordinates?
The radius is the square root of x squared plus y squared, and the angle is the arctangent of y over x, taken in the right quadrant.
What is the difference between cylindrical and spherical coordinates?
Cylindrical coordinates use a distance from the z axis, an angle and a height. Spherical coordinates use a distance from the origin and two angles.
Why do textbooks use different names for the angles?
Physics and mathematics books swap theta and phi. Check which angle is measured down from the z axis in your course.
Related calculators
Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any calculus or mathematics text (polar, cylindrical and spherical coordinates).
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