Plane from 3 Points
Plane equation and normal vector from three points
About this calculator
The Plane from 3 Points Calculator finds the equation of the plane ax + by + cz = d passing through any three points in 3D space, along with the plane's normal vector and unit normal. It detects collinear points that don't define a plane. Use it for analytic geometry, computer graphics, vectors, and engineering.
How to use the Plane from 3 Points calculator
- Enter the measurements you know into the input fields.
- Read the computed properties — they update instantly as you type.
- Compare the diagram with your figure to confirm the setup is right.
- Copy the page URL to share the exact calculation.
Common examples
- P₁(0,0,0), P₂(1,0,0), P₃(0,1,0) → z = 0 (the xy-plane), normal (0, 0, 1)
- P₁(1,0,0), P₂(0,1,0), P₃(0,0,1) → x + y + z = 1, normal (1, 1, 1)
- P₁(0,0,2), P₂(2,0,2), P₃(0,2,2) → z = 2, normal (0, 0, 1)
- P₁(0,0,0), P₂(1,1,1), P₃(2,2,2) → collinear points, no unique plane
Frequently asked questions
How does the Plane from 3 Points calculator work?
Enter the known measurements of your figure and the calculator applies the standard geometric formulas to find the remaining properties, alongside a diagram that reflects your numbers. All values are computed with high-precision arithmetic.
When would I use the Plane from 3 Points calculator?
It helps with geometry homework, technical drawing, construction and DIY layout, and any situation where you know some measurements of a shape and need the rest.
How accurate are the results?
Formulas are evaluated at 30 significant digits of precision internally, including roots and trigonometric functions. Displayed values are rounded for readability, but the underlying computation does not accumulate floating-point drift.
Can I share a specific calculation with someone else?
Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same figure and results for whoever clicks it.