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Distance from a Point to a Plane — result sheet
Calculate the unsigned and signed Cartesian distance from a point to a plane given by ax + by + cz + d = 0.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: unsigned distance = |a x + b y + c z + d|/sqrt(a^2+b^2+c^2); signed distance = (a x + b y + c z + d)/sqrt(a^2+b^2+c^2).
The point-to-plane distance divides the absolute plane expression by the Euclidean length of the nonzero normal. The signed result keeps the side indicated by the entered normal orientation.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Plane a — The x coefficient of the plane normal.; minimum -1000000; maximum 1000000
- Plane b — The y coefficient of the plane normal.; minimum -1000000; maximum 1000000
- Plane c — The z coefficient of the plane normal.; minimum -1000000; maximum 1000000
- Plane d — The constant in ax + by + cz + d = 0.; minimum -1000000; maximum 1000000
- Point x — The x coordinate of the point.; minimum -1000000; maximum 1000000
- Point y — The y coordinate of the point.; minimum -1000000; maximum 1000000
- Point z — The z coordinate of the point.; minimum -1000000; maximum 1000000
Worked example
| Input | Value |
|---|---|
| Plane a | 1 |
| Plane b | 2 |
| Plane c | 3 |
| Plane d | -6 |
| Point x | 2 |
| Point y | 1 |
| Point z | 1 |
The unsigned distance is 1/sqrt(14), approximately 0.267261, and the signed distance is positive.
Assumptions and limits
- The plane coefficients and point coordinates use one Cartesian 3D coordinate system, and (a,b,c) is nonzero.
- Unsigned distance is the shortest Euclidean distance; signed distance uses the entered normal orientation and does not infer physical units or a closest-point coordinate.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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