Spherical to Cartesian Calculator

Calculator Inputs

Radius ρ

Theta θ

Second Angle

Angle Unit

Second Angle Convention

Decimal Places

Formula Used

For the common polar convention, the second angle is measured from the positive z axis.

x = ρ sin(φ) cos(θ)

y = ρ sin(φ) sin(θ)

z = ρ cos(φ)

For the elevation convention, the second angle is measured from the xy plane.

x = ρ cos(α) cos(θ)

y = ρ cos(α) sin(θ)

z = ρ sin(α)

When degrees are selected, the calculator converts each angle into radians first. The conversion is radians = degrees × π / 180.

How to Use This Calculator

Enter the radius from the origin to the point.
Enter theta, the rotation angle in the xy plane.
Enter the second angle as either polar angle or elevation angle.
Select degrees or radians.
Choose decimal places for the final output.
Press the calculate button.
Review x, y, z, magnitude check, and normalized vector values.
Download the result as CSV or PDF when needed.

Example Data Table

Radius ρ	Theta θ	Second Angle	Convention	x	y	z
5	60°	45°	Polar	1.7678	3.0619	3.5355
10	30°	20°	Elevation	8.1379	4.6985	3.4202
7	180°	90°	Polar	-7.0000	0.0000	0.0000

Advanced Spherical Conversion Guide

Spherical coordinates describe a point by distance and direction. They are useful when motion, fields, or geometry radiates from one center. Engineers use them for antennas, scanners, optics, and three dimensional modeling. Students use them for vectors, integrals, and coordinate proofs.

What Each Input Means

The radius is the straight line distance from the origin. It is often named r or rho. The angle theta usually turns around the z axis. It measures direction inside the xy plane. The second angle depends on the selected convention. A polar angle starts at the positive z axis. An elevation angle starts at the xy plane.

Why Convention Matters

Many mistakes happen because books and software use different angle meanings. Physics texts often use polar angle phi. Navigation and mapping tools often use elevation. Both systems are valid. They only place the second angle in a different reference plane. This calculator makes the choice explicit. That helps prevent swapped sine and cosine terms.

Interpreting the Result

The x value shows movement along the horizontal x axis. The y value shows movement along the horizontal y axis. The z value shows height above or below the xy plane. A positive z means the point is above the plane. A negative z means it is below the plane. The magnitude check should match the radius after rounding.

Practical Uses

Spherical conversion helps when a problem begins with distance, direction, and tilt. A drone location can be described by range, bearing, and elevation. A camera ray can be described by length and two angles. A vector field can be sampled on shells. After conversion, the cartesian values are easier to plot, store, or compare.

Accuracy Tips

Use consistent angle units. Do not mix degrees and radians. Enter the radius as a nonnegative value. Choose enough decimals for your task. Higher precision is helpful for simulation. Lower precision is easier for reports. When angles are near quadrant boundaries, small rounding changes may flip a tiny value from positive to negative. Treat near zero outputs carefully.

Exporting Results

The CSV file supports spreadsheets. The PDF file supports quick sharing. Keep both with project notes, so every coordinate decision remains clearly traceable and auditable later.

FAQs

What is a spherical coordinate?

It is a three dimensional coordinate format using radius and two angles. The radius gives distance from the origin. The angles describe direction around and above the coordinate planes.

What does theta mean?

Theta is usually the azimuth angle. It rotates around the z axis inside the xy plane. It controls how the x and y values are split.

What is the polar angle?

The polar angle is measured downward from the positive z axis. In many math and physics texts, it is written as phi.

What is the elevation angle?

The elevation angle is measured upward or downward from the xy plane. It is common in navigation, surveying, radar, and practical direction problems.

Can I use radians?

Yes. Select radians from the angle unit field. When degrees are selected, the calculator converts the angles into radians before applying trigonometric formulas.

Why is the magnitude check shown?

The magnitude check confirms the distance of the cartesian point from the origin. It should match the entered radius, except for small rounding differences.

What happens when radius is zero?

All cartesian coordinates become zero. The point is exactly at the origin, so the direction angles no longer change the final location.

Why do tiny values appear near zero?

Trigonometric calculations can create very small decimal leftovers. They often appear when an exact result should be zero. Rounding usually removes them.