Spherical Coordinates Calculator

Convert between Cartesian (x,y,z) and Spherical (r, θ, φ) coordinates

Select your conversion type, input the coordinates, and get instant, accurate results. This tool is essential for applications in physics, engineering, and mathematics.

Examples

Click on an example to load it into the calculator.

Convert Cartesian Point

cartesianToSpherical

Convert a standard Cartesian point to its spherical equivalent.

x: 3

y: 4

z: 5

Point on Z-Axis

cartesianToSpherical

Convert a point lying directly on the Z-axis.

x: 0

y: 0

z: 10

Convert Spherical Point

sphericalToCartesian

Convert a standard spherical point to its Cartesian equivalent.

r: 10

θ: 60°

φ: 45°

Point on XY-Plane

sphericalToCartesian

Convert a spherical point where the polar angle is 90 degrees.

r: 5

θ: 90°

φ: 30°

Other Titles
Understanding Spherical Coordinates: A Comprehensive Guide
Explore the fundamentals of the spherical coordinate system and learn how to seamlessly convert between Cartesian and spherical representations.

What are Spherical Coordinates? Core Concepts

  • A 3D system using distance and two angles to define a point
  • Comprises radius (r), polar angle (θ), and azimuthal angle (φ)
  • Ideal for representing spheres, cones, and other symmetrical shapes
The spherical coordinate system is a three-dimensional framework for locating points in space using a combination of one distance and two angles. Unlike the Cartesian system's (x, y, z) linear axes, the spherical system offers a more natural way to describe objects with spherical symmetry.
The Three Components
1. Radius (r): Also known as the radial distance, 'r' is the distance from the origin (0,0,0) to the point. It is always non-negative (r ≥ 0).
2. Polar Angle (θ): Often called the inclination or colatitude, 'θ' (theta) is the angle measured from the positive Z-axis to the point's vector. Its value ranges from 0° to 180° (or 0 to π radians).
3. Azimuthal Angle (φ): Also known as the azimuth, 'φ' (phi) is the angle measured from the positive X-axis in the XY-plane to the projection of the point's vector onto that plane. It ranges from 0° to 360° (or 0 to 2π radians).

Key Coordinate Examples

  • Origin: (r=0, θ=undefined, φ=undefined)
  • Point on +Z axis: (r=k, θ=0°, φ=undefined)
  • Point on +X axis: (r=k, θ=90°, φ=0°)
  • Point on +Y axis: (r=k, θ=90°, φ=90°)

Step-by-Step Guide to Using the Calculator

  • Select your desired conversion direction
  • Input the known coordinates and choose your angle unit
  • Interpret the calculated results accurately
Our calculator simplifies the conversion process into a few easy steps, ensuring you get precise results quickly.
1. Choose Conversion Type
Start by selecting whether you want to convert from 'Cartesian to Spherical' or 'Spherical to Cartesian' from the dropdown menu. The input fields will adjust accordingly.
2. Select Angle Unit
Choose 'Degrees' or 'Radians' from the angle unit selector. This choice affects both how you input angles (for Spherical to Cartesian) and how the results are displayed (for Cartesian to Spherical).
3. Enter Your Coordinates
  • For Cartesian to Spherical: Fill in the x, y, and z input fields.
  • For Spherical to Cartesian: Fill in the r (radius), θ (polar angle), and φ (azimuthal angle) fields, ensuring the angles match your selected unit.
4. Calculate and View Results
Click the 'Calculate' button. The converted coordinates will appear in the 'Result' section. You can use the 'Reset' button to clear all fields and start a new calculation.

Practical Usage Examples

  • Cartesian to Spherical: Input x=1, y=1, z=1 with Degrees -> r=1.732, θ=54.74°, φ=45°
  • Spherical to Cartesian: Input r=1, θ=90, φ=90 with Degrees -> x=0, y=1, z=0

Mathematical Derivation and Formulas

  • The conversion formulas are derived from trigonometry
  • Understanding the relationship between the two systems
  • Key equations for both conversion directions
The conversion between Cartesian and spherical coordinates is rooted in right-triangle trigonometry. By projecting the point onto the XY-plane, we can form two right triangles that relate (x, y, z) to (r, θ, φ).
Cartesian to Spherical Conversion
Given a point (x, y, z):
  • Radius (r) is found using the 3D Pythagorean theorem: r = √(x² + y² + z²)
  • Polar Angle (θ) is found using the arccosine of the z-coordinate divided by the radius: θ = arccos(z / r)
  • Azimuthal Angle (φ) is found using the arctangent of y over x: φ = arctan(y / x). Care must be taken to place the angle in the correct quadrant, often using an atan2(y, x) function.
Spherical to Cartesian Conversion
Given a point (r, θ, φ):
  • x-coordinate: x = r * sin(θ) * cos(φ)
  • y-coordinate: y = r * sin(θ) * sin(φ)
  • z-coordinate: z = r * cos(θ)

Formula Application

  • Example: (x=1, y=√3, z=2). r = √(1² + (√3)² + 2²) = √8 = 2√2. θ = arccos(2 / 2√2) = 45°. φ = arctan(√3 / 1) = 60°.
  • Example: (r=4, θ=30°, φ=60°). x = 4 * sin(30) * cos(60) = 4 * 0.5 * 0.5 = 1. y = 4 * sin(30) * sin(60) = 4 * 0.5 * (√3/2) = √3. z = 4 * cos(30) = 4 * (√3/2) = 2√3.

Real-World Applications of Spherical Coordinates

  • Used extensively in physics, engineering, and geography
  • Simplifies problems with spherical symmetry
  • Fundamental to GPS, astronomy, and 3D modeling
The spherical coordinate system is not just a mathematical curiosity; it is a powerful tool used across numerous scientific and technical fields.
Physics and Engineering
  • Electromagnetism: Describing electromagnetic fields and radiation patterns from antennas is far simpler in spherical coordinates.
  • Quantum Mechanics: The wave functions of electrons in atoms (atomic orbitals) are expressed in spherical coordinates.
  • Fluid Dynamics: Analyzing fluid flow around a spherical object.
Geography and Astronomy
  • Global Positioning System (GPS): The Earth's geographic coordinate system (latitude and longitude) is a direct application of spherical coordinates. Latitude corresponds to the polar angle (θ), and longitude to the azimuthal angle (φ).
  • Celestial Mapping: Astronomers use celestial coordinate systems, which are analogous to spherical coordinates, to map the positions of stars and galaxies.
Computer Graphics and Robotics
  • 3D Modeling: Used to model spherical objects and create lighting effects that radiate from a point source.
  • Robotics: Defining the position and orientation of robotic arms that have rotational joints.

Industry Examples

  • Designing a satellite dish to focus signals.
  • Calculating the gravitational field of a planet.
  • Plotting the flight path of an intercontinental airplane.

Common Misconceptions and Special Cases

  • Clarifying the conventions for angles θ and φ
  • Handling undefined angles at the origin and Z-axis
  • Distinguishing between mathematical and physics conventions
While powerful, the spherical coordinate system has nuances that can lead to confusion. Understanding these is key to using it correctly.
The Origin (0,0,0)
At the Cartesian origin, the radius r is 0. In this case, both the polar angle θ and the azimuthal angle φ are undefined because there is no unique vector direction from the origin to itself.
Points on the Z-Axis
For any point on the Z-axis (where x=0 and y=0), the azimuthal angle φ is undefined. This is because the projection of the point onto the XY-plane is the origin, and there is no clear angle from the X-axis. By convention, it is often set to 0.
Math vs. Physics Convention
It's crucial to be aware of different conventions. This calculator uses the standard mathematical convention where θ is the inclination from the Z-axis and φ is the azimuth from the X-axis. In physics, the roles of θ and φ are often swapped, where θ becomes the azimuth and φ the inclination. Always check the convention being used in your context.

Edge Case Scenarios

  • Input: (x=0, y=0, z=5) -> r=5, θ=0°, φ is undefined (often shown as 0°).
  • Input: (x=0, y=0, z=-5) -> r=5, θ=180°, φ is undefined (often shown as 0°).
  • Be aware: Some systems define θ from the XY-plane, not the Z-axis.