Spherical to Rectangular Calculator

Understanding Spherical Coordinates

Spherical coordinates describe a point by distance and two angles. The radius tells how far the point sits from the origin. The azimuth angle turns around the horizontal plane. The second angle controls height. Many courses call that angle phi. Some tools call it inclination. Others use elevation instead. This calculator supports both styles.

The rectangular system uses x, y, and z components. These components describe movement along three perpendicular axes. They are easier to plot on grids. A spherical value is compact. A rectangular value is direct. Converting between them helps you move between geometry and computation.

Formula Used

For the inclination convention, the angle phi is measured from the positive z-axis. The formulas are x = r sin(phi) cos(theta), y = r sin(phi) sin(theta), and z = r cos(phi). This style is common in calculus, physics, and vector analysis.

For the elevation convention, the elevation angle is measured above the xy-plane. The formulas are x = r cos(elevation) cos(theta), y = r cos(elevation) sin(theta), and z = r sin(elevation). This style often appears in navigation, surveying, graphics, and camera work.

The calculator converts degree entries into radians before calculation. Radian entries are used directly. The check should match the original radius, except for rounding.

How to Use This Calculator

Enter the radius first. Use a nonnegative value. Then enter the azimuth angle. This angle usually starts on the positive x-axis. It rotates toward the positive y-axis. Choose degrees or radians. Next, enter the second angle. Select the convention that matches your problem statement.

Pick the decimal precision you need. Smaller precision is cleaner for reports. Larger precision is better for engineering work. Press the calculate button. The result appears above the form. Review the x, y, and z values. Check the xy projection when you need the flat distance from the z-axis.

Working With Angle Conventions

Angle convention is the most common source of mistakes. In inclination mode, zero degrees points along the z-axis. Ninety degrees lies on the xy-plane. In elevation mode, zero degrees lies on the xy-plane. Ninety degrees points upward. These definitions are different. They can produce very different z values.

When a textbook uses rho, theta, and phi, it often means inclination. When a drawing shows an angle above the ground plane, it usually means elevation. Always read the diagram carefully. Convert the convention first when needed.

Accuracy and Practical Uses

This conversion is useful in 3D design, robotics, antenna direction, physics, games, and mapping. It can place a point from a distance and direction. It can also turn polar-like measurements into coordinates for code. Rounding can slightly change the returned radius. Use more decimals for careful comparison.

Use consistent units for distance. Angles need their own unit setting. A radius in meters returns x, y, and z in meters. A radius in feet returns components in feet. Reliable coordinate conversion supports better graphs, models, and decisions.