Calculate radial distance from the origin using x, y and z coordinates, or the distance between two points, with direction angles and unit conversion.
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Radial Distance Formula
The calculator uses Euclidean distance, its inverse relationships and polar or spherical coordinate conversions for the selected mode.
From origin (2D):
Between two points (2D):
3D radius from origin:
- r — radial distance from the origin in 2D
- d — straight-line distance between two points
- ρ — radial distance from the origin in 3D
- x, y, z — Cartesian coordinates
- x1, y1, x2, y2 — coordinates of the two points
All coordinate and radius inputs share the selected input length unit. The output length unit can convert physical values; generic units have no physical conversion. Radial distance is always non-negative. All direction angles are undefined at the origin or for coincident points. In 3D, azimuth is also undefined on the Z axis, while polar angle and elevation remain defined away from the origin. A missing coordinate is ±√(r² − the sum of known coordinate squares): zero gives one solution, and a negative radicand gives no real solution.
Reference Tables
Quick checks for common coordinates and the matching radial distance.
| Point (x, y) | r = √(x² + y²) | Angle θ |
|---|---|---|
| (3, 4) | 5 | 53.13° |
| (1, 1) | 1.4142 | 45° |
| (5, 12) | 13 | 67.38° |
| (8, 15) | 17 | 61.93° |
| (-6, 8) | 10 | 126.87° |
| (0, 7) | 7 | 90° |
Unit conversion factors for translating a calculated radial distance.
| From | To meters | To feet |
|---|---|---|
| 1 mm | 0.001 | 0.00328 |
| 1 cm | 0.01 | 0.0328 |
| 1 in | 0.0254 | 0.0833 |
| 1 ft | 0.3048 | 1 |
| 1 yd | 0.9144 | 3 |
| 1 mi | 1609.344 | 5280 |
Example and FAQ
Example: Find the radial distance from the origin to the point (9, 12).
r = √(9² + 12²) = √(81 + 144) = √225 = 15. The angle is θ = atan2(12, 9) ≈ 53.13°, so the polar coordinates are (15, 53.13°).
Is radial distance the same as the magnitude of a vector? Yes. For a position vector starting at the origin, the radial distance equals the vector magnitude.
Can radial distance be negative? No. It is a length, so it is always zero or positive. A negative sign in polar form flips the direction, not the distance.
What is the difference between azimuth and elevation in 3D? Azimuth is the angle in the XY plane measured from the positive X-axis. Elevation is the angle of the point above or below the XY plane.
Does the unit affect the answer? The selected input unit applies to every coordinate and radius entry. Converting a physical result to a different output unit changes its numerical value. For example, 5 meters equals 500 centimeters. Generic units cannot be converted to meters or feet without a physical scale.
