Coordinate Converter
Enter radians directly, or use expressions such as pi/2, 2pi/3, or 1.25.
Formula Used
For the common polar-angle convention, the spherical point is written as (ρ, θ, φ). Here, ρ is radius, θ is azimuth, and φ is measured down from the positive z-axis.
x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ), and z = ρ cos(φ).
For the elevation convention, the third angle is α, measured up from the xy-plane. Then x = ρ cos(α) cos(θ), y = ρ cos(α) sin(θ), and z = ρ sin(α).
How to Use This Calculator
Enter the radius first. Then enter the azimuth angle in radians. Add the third angle using the convention selected in the dropdown. Use decimal values or expressions with pi. Select precision and rounding. Add origin shifts only when your coordinate system starts from another point. Press the calculate button to view the rectangular coordinates above the form.
Understanding the Conversion
Spherical coordinates describe a point by distance and two angles. Rectangular coordinates describe the same point with three signed distances. The two systems are useful in different settings. Spherical form is compact for points on spheres, cones, waves, fields, and rotations. Rectangular form is easier for graphing, measuring components, and using vectors in many equations.
This calculator focuses on radian input. Radians match the natural measure used by sine and cosine functions. A full turn is 2π radians. A right angle is π/2 radians. This is why the input box accepts expressions such as pi/4 or 3pi/2. The parser converts them before applying the trigonometric formulas.
The most important choice is the third angle convention. In many mathematics courses, φ is the polar angle from the positive z-axis. In some engineering and mapping work, the third angle is elevation from the xy-plane. These two angles are complementary. Confusing them can move the point to a different height. The calculator includes both choices to prevent that mistake.
The azimuth angle θ always turns around the xy-plane from the positive x-axis. Its cosine controls the x share. Its sine controls the y share. The third angle controls how much of the radius stays in the xy-plane and how much becomes vertical height. Together, these pieces split one radial distance into three rectangular components.
Precision controls are helpful because trigonometric calculations often produce tiny floating values. For example, a theoretical zero may appear as a very small decimal. The cleanup option can display that value as zero. Rounding choices let you prepare answers for homework, reports, code checks, or design notes. The radius check shows whether the converted components still match the original distance.
Origin shifts add another advanced option. Standard formulas assume the sphere is centered at the origin. If your sphere is centered at another point, enter x₀, y₀, and z₀. The converter first calculates the normal rectangular components. It then adds the offsets. This gives coordinates in the shifted reference frame without changing the angle formulas.
Negative radius is usually not used in standard spherical coordinates. Some algebra problems may allow it as a signed radial value. The calculator keeps it disabled by default. Enable it only when your source specifically permits signed radius. For most geometry, physics, and vector problems, radius should be zero or positive.
A good workflow is simple. Confirm the angle convention. Enter radius and angles. Choose output precision. Review the ordered triplet. Then compare the radius check with the original radius. If the difference is near zero, the conversion is consistent. If the values look wrong, the angle convention is usually the first thing to inspect. This careful process keeps coordinate conversion clear and reliable. It also helps when copying values into spreadsheet cells, plotting tools, or simulation inputs, because every component appears with the same selected precision and unit label clearly.
FAQs
What does spherical to rectangular conversion mean?
It changes a point from radius and two angles into x, y, and z coordinates. The point stays the same. Only the coordinate description changes.
Does this calculator use radians?
Yes. All angle inputs are treated as radians. You can enter decimal radians or expressions such as pi/6, pi/2, and 2pi/3.
Which angle convention should I choose?
Choose polar angle when the third angle is measured from the positive z-axis. Choose elevation when it is measured upward from the xy-plane.
What is θ in spherical coordinates?
θ is the azimuth angle. It rotates around the xy-plane from the positive x-axis toward the positive y-axis.
What is φ in the standard formula?
φ is commonly the polar angle measured from the positive z-axis. It controls the vertical part and the xy-plane projection.
Can I enter pi expressions?
Yes. The calculator accepts pi, π, fractions, parentheses, and basic operators. Examples include pi/4, 3*pi/2, and (2*pi)/3.
Why do I see a very small value instead of zero?
Floating calculations can create tiny rounding artifacts. Enable near-zero cleanup to display values below your selected limit as zero.
What does origin shift do?
Origin shift adds x₀, y₀, and z₀ after conversion. Use it when the spherical coordinate system is centered away from the main origin.
Can radius be negative?
Standard spherical coordinates use a nonnegative radius. The negative radius option is included for special algebra cases where signed radius is allowed.
How can I verify the answer?
Check the radius test. The distance from the calculated components back to the origin should match the entered radius, apart from rounding.
Can I use this for vectors?
Yes. A position vector can be converted from spherical components to rectangular components when the radius and angles describe its direction.