Free · Coordinate geometry
Cartesian and Cylindrical Coordinate Pair
Convert rectangular and cylindrical coordinates in both panels, preserving quadrant-aware azimuth and signed heights.
- Formula & worked example
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
The point (−3,3,−2) has radius √18, azimuth135° and height−2.
r = sqrt(x²+y²); θ = atan2(y,x); x = r cos θ; y = r sin θ; z stays the same.
Worked example
Enter these known values and leave the other values blank.
- x
- -3
- y
- 3
- z
- -2
- Radius (ρ)
- 4.243
- θ
- 135 deg
- z
- -2
Assumptions and limitations
- The two panels convert in opposite directions independently. Radius is nonnegative; height may be signed. All coordinates must use the same length unit.
- Rectangular-to-cylindrical azimuth uses the principal interval greater than−180° and at most180°. Coterminal angles represent the same ray.
- At x=y=0, radius is zero and height is retained, but azimuth has no unique value. The angle remains blank because it is undefined; this is not an invalid point. Radius and height remain meaningful.
- The captured internal square-root inverses choose their nonnegative branch; output-only controls remain read-only. Use the cylindrical-input panel for a complete radius/azimuth-to-Cartesian conversion.
- Trigonometric evaluation supports input angles with magnitude at most 1,000,000 radians (about 159,155 full turns). This computational range is not a physical limit. Reduce larger angles to a coterminal angle before entering them; all nine angle units remain available.
Common questions
Why does a point with negative x need atan2?
The signs of both x and y determine the quadrant. Dividing y by x discards that quadrant and is undefined when x is zero.
Can radius be zero?
Yes. Every azimuth then describes the same point on the z axis; a unique angle cannot be recovered from x=y=0.
References
Use the full-quadrant azimuth on (-pi,pi]; the z axis has no unique azimuth. The original atan(y/x) loses the sign quadrant and divides by zero on the y axis. The zero-radius azimuth expression remains undefined, with finite-result validation rather than an invented zero angle. All ten other variants are retained. Trigonometric variants are guarded before evaluation at an explicitly disclosed computational limit of |θ| ≤ 1,000,000 radians; this is not a physical angle restriction. Outside it, reduce the angle to a coterminal value. The guard prevents unsupported argument reduction and blank unresolved results, with an actionable message. The two forward consistency identities use the same lazy numerical-angle guard, so previous finite outputs cannot trigger unsupported trigonometry after a boundary-to-invalid transition. Outside that numerical domain the identity yields no finite comparison; the range error remains actionable. Evaluate the captured nonnegative radius requirement by sign instead of a fuzzy comparison with zero. This retains the mathematical domain while handling extremely small signed values correctly; it adds no physical magnitude cap.