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Plane from Three Points Calculator

Enter the x, y, z coordinates of three non-collinear points.

How it is calculated

Build two edge vectors from point A: AB = B − A and AC = C − A. Their cross product n = AB × AC = (nx, ny, nz) is perpendicular to the plane, so A = nx, B = ny, C = nz in Ax + By + Cz + D = 0. D follows by substituting point A: D = −(nx·ax + ny·ay + nz·az).

The nine inputs are the x, y, z coordinates of A, B and C in any consistent unit. The coefficients are printed to four decimals and are not divided by the length of the normal, so they are not a unit normal and can be scaled by any non-zero factor to describe the same plane. If the three points lie on one straight line (or two coincide) the cross product is zero and the tool gives no result.

Worked example

Take A = (1, 0, 0), B = (0, 2, 0) and C = (0, 0, 3). Then AB = (−1, 2, 0) and AC = (−1, 0, 3). The cross product is (2·3 − 0·0, 0·(−1) − (−1)·3, (−1)·0 − 2·(−1)) = (6, 3, 2). Using point A, D = −(6·1 + 3·0 + 2·0) = −6. The plane is 6x + 3y + 2z − 6 = 0; the tool shows A = 6.0000, B = 3.0000, C = 2.0000 and D = −6.0000. B and C satisfy it too: 3·2 − 6 = 0 and 2·3 − 6 = 0.

Try this example →

Common mistakes

  • Entering three points on one line, such as (1,1,1), (2,2,2) and (3,3,3). They do not define a single plane, so the tool gives no result.
  • Taking the printed A, B, C as a unit normal. They are not normalised; divide by the vector length yourself if you need a unit normal.
  • Expecting the signs to be fixed. Swapping the order of the points flips the normal and the sign of D, though the plane itself stays the same.
  • Mixing units between coordinates, for example x in metres and z in centimetres. All nine numbers must share a unit.
  • Typing a coordinate into the wrong box, such as by into bx, which silently describes a different plane.

About this calculator

Given three non-collinear points, the plane's normal vector is found by taking the cross product of two edge vectors formed from the points: n = (P₂ − P₁) × (P₃ − P₁). That normal vector's components become the A, B, C coefficients in the plane equation Ax + By + Cz + D = 0, and D is solved by plugging any one of the three points back into the equation.

3D graphics programmers use this to compute surface normals for lighting and collision calculations, CAD and BIM engineers use it to define reference planes and cutting sections from measured points on a model, and surveyors and geodesists use the same cross-product method to fit a best-fit ground plane through three elevation-surveyed points on a site.

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Comments (1)

  • Editorial team

    Point order decides which way the normal faces, which matters for lighting and collision work. With A at the origin, B at (1,0,0) and C at (0,1,0), the C coefficient is 1.0000; swap B and C and it becomes -1.0000, though the plane is identical. If your engine expects outward-facing normals, order the points consistently around each face. What are you building these planes for?

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