Equation Of A Plane Calculator

Work out equation of a plane instantly with clear inputs, formula shown and shareable results.

About Equation Of A Plane Calculator

Two edge vectors AB and AC lie in the plane, so their cross product is a normal vector (a, b, c). Substituting point A then fixes d in ax + by + cz + d = 0. Half the normal's magnitude is the area of triangle ABC.

Formulas

Normal from three points

n = (B − A) × (C − A)

Plane equation

a(x − x_A) + b(y − y_A) + c(z − z_A) = 0

Worked example

With the default values below, Equation Of A Plane Calculator returns Equation of the plane of 6x + 3y + 2z - 6 = 0. Change any input above and every figure here is recalculated in your browser.

Inputs used

Point A (x, y, z): 1, 0, 0
Point B (x, y, z): 0, 2, 0
Point C (x, y, z): 0, 0, 3

Results

Equation of the plane: 6x + 3y + 2z - 6 = 0
Normal vector: (6, 3, 2)
Constant d: -6.0000
Area of triangle ABC: 3.5000

Step by step

1Values used
2Normal from three points
3Plane equation
4Equation of the plane= 6x + 3y + 2z - 6 = 0
5Normal vector= (6, 3, 2)
6Constant d= -6.0000
7Area of triangle ABC= 3.5000

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For (1,0,0), (0,2,0), (0,0,3)?

The normal is (6, 3, 2) and the plane is 6x + 3y + 2z − 6 = 0, matching the intercept form x/1 + y/2 + z/3 = 1.

Why must the points be non-collinear?

Collinear points lie on infinitely many planes, so the cross product is the zero vector and no unique answer exists.