Plane Equation Calculator

Calculator Input

Input mode

Decimal precision

Graph range

Three Points

P1 x

P1 y

P1 z

P2 x

P2 y

P2 z

P3 x

P3 y

P3 z

Point and Normal Vector

Point x

Point y

Point z

Normal a

Normal b

Normal c

General Coefficients

a

b

c

d

Optional Test Point

Test x

Test y

Test z

Reset

Example Data Table

Mode	Input Data	Derived Plane	Key Notes
Three points	P1 = (1,0,0), P2 = (0,1,0), P3 = (0,0,1)	x + y + z - 1 = 0	All intercepts are 1.
Point and normal	P = (2,-1,3), n = (4,2,-5)	4x + 2y - 5z + 9 = 0	Normal vector controls plane orientation.
Coefficients	a = 3, b = -2, c = 1, d = -6	3x - 2y + z - 6 = 0	Useful when the general form is known.

Formula Used

General plane form
ax + by + cz + d = 0

From a point and a normal vector
a(x - x₁) + b(y - y₁) + c(z - z₁) = 0

From three points
n = (P₂ - P₁) × (P₃ - P₁)
n · (r - r₀) = 0

Distance from point (x₀, y₀, z₀) to plane
|ax₀ + by₀ + cz₀ + d| / √(a² + b² + c²)

Axis intercepts
x = -d/a, y = -d/b, z = -d/c when the related coefficient is non-zero.

How to Use This Calculator

Select how you want to define the plane.
Enter either three points, one point with a normal vector, or the general coefficients.
Optional: add a test point to check whether it lies on the plane.
Choose decimal precision and an optional graph range.
Press Calculate Plane to see the equation, traces, intercepts, distances, and graph.
Use the CSV or PDF buttons to export the result summary.

FAQs

1. What inputs can this calculator use?

It supports three common methods: three non-collinear points, one point with a normal vector, or direct coefficients in general plane form.

2. Why do three points sometimes fail?

A unique plane needs three distinct, non-collinear points. If the points lie on one straight line, infinitely many planes pass through them.

3. Why is the intercept form sometimes unavailable?

Intercept form needs finite, non-zero axis intercepts. It is unavailable when the plane passes through the origin or is parallel to an axis.

4. What does the normal vector mean?

The normal vector points straight out from the plane. It determines the plane orientation and appears directly in the point-normal equation.

5. How is point-to-plane distance computed?

The calculator substitutes the point into the plane equation, takes the absolute value, and divides by the normal vector magnitude.

6. What is the substitution value for a test point?

It is the result of ax + by + cz + d. Zero means the point lies on the plane. Positive and negative values show opposite sides.

7. Can the same plane have different equations?

Yes. Multiplying every coefficient by the same non-zero constant gives an equivalent equation for exactly the same plane.

8. How should I choose the graph range?

Use a range slightly larger than your points or intercepts. Smaller ranges improve readability when your coordinates are close together.