Example data table
| # | Point (x₀, y₀, z₀) | Plane (a, b, c, d) | Distance |
|---|---|---|---|
| 1 | (2, -1, 4) | (1, -2, 3, 6) | 2.1381 |
| 2 | (0, 0, 5) | (0, 0, 1, -2) | 3 |
| 3 | (-3, 2, 1) | (2, 1, -2, 4) | 2.3333 |
Numbers are illustrative; your results depend on selected precision.
Formula used
For a plane a x + b y + c z + d = 0 and a point (x₀, y₀, z₀), the perpendicular distance is:
If you enter the plane using three points or normal form, the calculator converts it into (a, b, c, d) first.
How to use this calculator
- Select a plane input mode that matches your problem.
- Enter the 3D point coordinates (x₀, y₀, z₀).
- Fill plane inputs: coefficients, three points, or normal and point.
- Pick decimal precision and optional unit label.
- Press Calculate to see results above the form.
- Use CSV or PDF buttons to export your saved results.
Recent calculations
No calculations saved yet. Run one to populate this list.
FAQs
1) What does the distance represent?
It is the shortest, perpendicular distance from the point to the plane. The value is always non‑negative and uses the plane’s normal direction.
2) Why do you take the absolute value in the formula?
The numerator can be positive or negative depending on which side of the plane the point lies. Absolute value converts that signed separation into a true distance.
3) What if a, b, and c are all zero?
Then the equation does not describe a plane because the normal vector has zero length. The calculator flags this as invalid and won’t compute a distance.
4) Can I enter a plane using three points?
Yes. The calculator forms two direction vectors from the points, takes their cross product to get a normal, then builds the plane coefficients automatically.
5) What causes “three points do not form a unique plane”?
If the points are identical, collinear, or nearly collinear, the cross product becomes zero. That means there is no single plane passing through all three points.
6) Does scaling the plane coefficients change the distance?
No. Multiplying (a, b, c, d) by any non‑zero constant scales numerator and denominator equally, so the computed distance stays the same.
7) Can I see which side of the plane my point is on?
Yes. The “signed numerator” value keeps its sign. Positive and negative indicate opposite sides relative to the plane’s normal direction.
8) What do the CSV and PDF downloads include?
They include your session’s saved calculations with timestamps, point, plane, intermediate values, and the final distance. The PDF summarizes the latest result and shows a short history list.