Cartesian to Spherical Polar Coordinates Calculator

Calculator Inputs

Formula Used

The calculator uses the standard physics convention for spherical polar coordinates.

Radius: r = √(x² + y² + z²)

XY projection: ρ = √(x² + y²)

Polar angle: θ = arccos(z / r)

Azimuth: φ = atan2(y, x)

Elevation: α = arcsin(z / r)

Reverse check: x = r sin(θ) cos(φ), y = r sin(θ) sin(φ), z = r cos(θ)

How to Use This Calculator

Enter the x, y, and z coordinates of the point.
Select degrees or radians for the angle outputs.
Choose a positive full-turn azimuth or signed half-turn azimuth.
Set the decimal precision for rounded results.
Press the convert button to show the result above the form.
Use the CSV or PDF button to save the computed report.

Example Data Table

x	y	z	r	θ degrees	φ degrees	Use case
3	4	5	7.071068	45.000000	53.130102	General vector point
-2	6	3	7.000000	64.623067	108.434949	Quadrant check
0	0	8	8.000000	0.000000	Undefined	Z-axis point

Cartesian to Spherical Polar Coordinates Guide

Cartesian coordinates describe a point with x, y, and z distances from three perpendicular axes. Spherical polar coordinates describe the same point with a radius and two angles. This calculator helps you move between those views without manual trigonometry. It is useful in vectors, multivariable calculus, physics, robotics, surveying, and three dimensional geometry.

Why This Conversion Matters

Many shapes and fields are easier to study with spherical variables. A sphere, cone, radial force, antenna pattern, or orbital path often has a simpler equation after conversion. The radius shows distance from the origin. The polar angle shows tilt from the positive z axis. The azimuth shows rotation around the xy plane. These values make direction and distance easier to compare.

Advanced Calculator Features

The tool accepts positive, negative, decimal, and zero values. It reports radius, xy projection, polar angle, azimuth, elevation angle, direction cosines, and a reverse coordinate check. You can choose degrees or radians. You can also normalize the azimuth into a full positive turn or keep a signed angle. The result area appears before the form, so the answer is visible after submission.

Accuracy And Interpretation

The calculator uses atan2 for azimuth. This keeps the correct quadrant when x or y is negative. It uses arccos for the polar angle when the radius is not zero. At the origin, the radius is zero and the angles are not unique. In that special case, the page displays a helpful note instead of hiding the issue.

Practical Uses

Students can verify homework steps. Teachers can create examples for lessons. Engineers can check vector directions. Data teams can transform spatial measurements before plotting. The CSV download stores the main values for spreadsheets. The PDF download creates a compact report for printing or sharing.

Best Practice

Use consistent units for x, y, and z. Review the selected angle unit before copying results. For physics notation, theta usually means the polar angle from the positive z axis. Phi usually means the azimuth measured from the positive x axis in the xy plane. For clear records, note the coordinate convention beside every answer. Different textbooks may swap angle symbols, so labels matter more than symbol names during careful work.

FAQs

What are spherical polar coordinates?

They describe a point with radius, polar angle, and azimuth. The radius gives distance from the origin. The angles describe direction in three dimensional space.

What does theta mean here?

Theta means the polar angle measured from the positive z-axis. This is common in physics and many vector calculus courses.

What does phi mean here?

Phi means the azimuth angle in the xy plane. It is measured from the positive x-axis toward the positive y-axis.

Why does the calculator use atan2?

atan2 uses both x and y signs. That helps place the azimuth in the correct quadrant, even when one coordinate is negative.

What happens at the origin?

At the origin, the radius is zero. Direction is not unique, so both spherical angles are undefined. The page displays a note for this case.

Can I use radians instead of degrees?

Yes. Select radians in the angle unit field. The calculator will output theta, phi, and elevation in radians.

What is the reverse check?

The reverse check converts the spherical result back to x, y, and z. Small errors may appear because of decimal rounding.

Why is azimuth undefined on the z-axis?

When x and y are both zero, every horizontal direction is equivalent. The point has no unique rotation around the z-axis.