Math
Spherical Coordinates Calculator
Spherical coordinates describe a point by its distance ρ from the origin, an azimuth angle around the z-axis, and a polar angle measured down from the positive z-axis.
Math textbooks write these as (ρ, θ, φ) with θ the azimuth; physics and ISO 80000-2 write (r, θ, φ) with θ the polar angle. Choose your convention, enter a point in any system, and mathcheck converts it with the exact radicals, the quadrant-correct angles in degrees and radians, and every formula shown.
Try 1.4142136 for √2.
Try an example
Result
Converted point
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- ρ (distance from the origin)
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- Azimuth (θ math / φ physics), degrees
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- Azimuth, radians
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- Polar angle from +z (φ math / θ physics), degrees
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- Polar angle, radians
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- x
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More details (3 more)
- y
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- z
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- Cylindrical form
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What spherical coordinates are
A point in space can be located by how far it is from the origin (ρ) and which direction it lies in. The direction takes two angles: an azimuth measured around the z-axis from the positive x-axis, exactly like θ in polar coordinates, and a polar angle measured down from the positive z-axis. A polar angle of 0° points straight up the z-axis, 90° lies in the xy-plane, and 180° points straight down.
Math convention versus physics convention
The formulas are identical; only the letters differ, and that is the biggest source of errors. American calculus textbooks (including OpenStax) write (ρ, θ, φ) with θ as the azimuth and φ as the polar angle. Physics, engineering, and the ISO 80000-2 standard write (r, θ, φ) with θ as the polar angle and φ as the azimuth. The toggle above relabels the inputs and the answer so the letters match your course.
| Quantity | Math convention | Physics / ISO convention |
|---|---|---|
| Distance from the origin | ρ | r |
| Azimuth (around z, from +x) | θ | φ |
| Polar angle (down from +z) | φ | θ |
| Order written | (ρ, θ, φ) | (r, θ, φ) |
(1, 1, √2) to spherical
ρ = sqrt(1 + 1 + 2) = 2. Azimuth: (1, 1) is in Quadrant I with reference angle 45°, so the azimuth is 45° = π/4. Polar angle: arccos(√2 / 2) = 45° = π/4. Math form (2, π/4, π/4); physics form (2, π/4, π/4) happens to look the same here because both angles are equal.
(4, 45°, 60°) in the math convention to rectangular
x = 4 sin 60° cos 45° = 4(√3/2)(√2/2) = √6, y = 4 sin 60° sin 45° = √6, z = 4 cos 60° = 2. The point is (√6, √6, 2).
How to use this calculator
- Pick the naming convention your textbook uses so the letters θ and φ line up.
- Choose the conversion: rectangular to spherical, spherical to rectangular, or spherical to cylindrical.
- Enter the coordinates. Angles accept degrees, An angle unit where a full turn is 2π instead of 360 degrees., or π fractions like pi/3.
- Read the converted point, then the individual values: ρ, the azimuth and polar angle in degrees and radians, x, y, z, and the cylindrical form.
- Open Show the work for the A root expression written with the √ symbol, such as a square root., the One of the four regions the x- and y-axes cut the coordinate plane into. reasoning for the azimuth, the arccos step for the polar angle, and the exact trig values.
How to read the answer
ρ is always non-negative. The azimuth is reported in [0°, 360°) and the polar angle in [0°, 180°]; a polar angle above 90° means the point is below the xy-plane. Exact radicals appear when the angles are multiples of 30° or 45°; otherwise the calculator shows four-decimal values. The cylindrical form is included because r = ρ sin(polar) and z = ρ cos(polar) are the two legs of the right triangle that ρ closes.
Common mistakes and edge cases
- Mixing the math and physics conventions: swapping θ and φ moves the point to a completely different place.
- Measuring the polar angle from the xy-plane instead of from the +z axis.
- Using atan(y/x) for the azimuth without checking the quadrant of (x, y).
- Computing the polar angle with arctan(z / r) but forgetting that r = sqrt(x^2 + y^2) must be non-negative; arccos(z / ρ) avoids the ambiguity.
- Entering a negative ρ. Distances from the origin are never negative.
Worked examples
Rectangular to spherical, exact
(1, 1, √2)
Converted point
(ρ, θ, φ) = (2, 45°, 45°)
Point below the xy-plane
(0, 3, -4)
Converted point
(ρ, θ, φ) = (5, 90°, 143.1301°)
Physics convention swaps the angle order
(0, 3, -4) written as (r, θ, φ)
Converted point
(r, θ, φ) = (5, 143.1301°, 90°)
The origin
Converted point
(ρ, θ, φ) = (0, 0°, 0°)
Spherical to rectangular, exact radicals
(4, 45°, 60°) math convention
Converted point
(x, y, z) = (√6, √6, 2)
Spherical to rectangular with π fractions
(2, π, π/2): on the negative x-axis
Converted point
(x, y, z) = (-2, 0, 0)
Physics convention input
(r, θ, φ) = (4, 60°, 45°) is the same point as the math example above
Converted point
(x, y, z) = (√6, √6, 2)
Non-special angles
(3, 20°, 70°)
Converted point
(x, y, z) = (2.6491, 0.9642, 1.0261)
Spherical to cylindrical
(4, 45°, 60°)
Converted point
(r, θ, z) = (2√3, 45°, 2)
Negative ρ
Converted point
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Frequently asked questions
How do I convert rectangular to spherical coordinates?+
ρ = sqrt(x^2 + y^2 + z^2). The azimuth comes from the position of (x, y) in the xy-plane using a reference angle and the quadrant. The polar angle is arccos(z / ρ). For (1, 1, √2): ρ = 2, azimuth 45°, polar angle 45°.
What is the difference between the math and physics conventions?+
Only the letters. Math texts use (ρ, θ, φ) with θ as the azimuth; physics and ISO 80000-2 use (r, θ, φ) with θ as the polar angle from +z and φ as the azimuth. The calculator's toggle relabels everything so you can copy the answer in either style.
What range do the angles have?+
The azimuth runs over a full turn, [0°, 360°) or [0, 2π). The polar angle runs from 0° (the +z axis) to 180° (the -z axis); it never needs to exceed 180° because the azimuth handles the other half.
How do spherical and cylindrical coordinates relate?+
They share the azimuth. The cylindrical radius is r = ρ sin(polar) and the height is z = ρ cos(polar), so ρ is the hypotenuse of a right triangle with legs r and z.
Why is the polar angle measured from the z-axis instead of the xy-plane?+
Convention, chosen so that the volume element ρ^2 sin(φ) dρ dθ dφ and the formulas for x, y, z come out in their simplest form. Latitude on Earth is measured from the equator, so latitude = 90° - polar angle.
Which surfaces are simplest in spherical coordinates?+
A sphere centered at the origin is ρ = constant, a cone around the z-axis is φ = constant (polar angle), and a half-plane through the z-axis is θ = constant (azimuth).
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026