Free Distance from Point to Plane Calculator

Last updated October 10, 2026 · Report an error

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The shortest distance from a point to a plane is always measured along the perpendicular dropped from the point to the plane — any other path is longer. This calculation shows up in 3D computer graphics and game physics (checking whether a character or object has collided with a wall or floor), in CNC machining and 3D printing (verifying clearance between a tool path and a surface), and in structural and civil engineering when checking a point's offset from a design surface. Given the plane's general equation Ax+By+Cz+D=0 and any point's coordinates, the distance follows from one closed-form formula.

How it works

Enter the point's coordinates and the plane's A, B, C, D coefficients. The calculator plugs the point into the plane's equation to get Ax0+By0+Cz0+D (which is zero exactly when the point lies on the plane), takes the absolute value, and divides by √(A²+B²+C²) — the magnitude of the plane's normal vector. This scales the raw "how far off the equation is from zero" value into an actual physical distance.

Enter point — x0.
Enter point — y0.
Enter point — z0.
Enter plane — A (coefficient of x).
Enter plane — B (coefficient of y).
Enter plane — C (coefficient of z).
Enter plane — D (constant).
Click Calculate to see your results.

Examples

Point (1,2,3), plane 2x-y+2z-4=0

Numerator = |2(1)-2+2(3)-4| = |2-2+6-4| = |2| = 2. Denominator = √(2²+(-1)²+2²) = √9 = 3. Distance = 2/3 ≈ 0.667.

Point (0,0,0), plane x+y+z-6=0

Numerator = |0+0+0-6| = 6. Denominator = √(1+1+1) = √3 ≈ 1.732. Distance = 6/1.732 ≈ 3.464.

Common mistakes to avoid

Forgetting the absolute value in the numerator, which would let the "distance" come out negative.
Using the plane's D coefficient with the wrong sign — make sure your plane is written as Ax+By+Cz+D=0, not Ax+By+Cz=D (which would need D negated).
Forgetting to include all three normal-vector components (A, B, C) when computing the denominator, especially if one coefficient is zero.
Confusing this with the distance from a point to a line in 3D, which is a different (and more involved) calculation requiring a direction vector, not just a normal vector.

Frequently asked questions

The distance will be exactly 0, since the numerator (the plane equation evaluated at the point) is zero.

The vector (A, B, C) is the plane's normal vector, and its length isn't necessarily 1. Dividing by that length rescales the equation so the result is a true distance rather than an arbitrary multiple of it.

Yes — this is the direct 3D generalization. The 2D formula d=|Ax0+By0+C|/√(A²+B²) simply gains a z-term and a C-coefficient becomes D.

Yes — the raw signed value Ax0+By0+Cz0+D tells you which side of the plane the point is on (positive or negative), which is useful in some graphics and collision algorithms, but the "distance" itself is always reported as a non-negative value.

That's not a valid plane — the normal vector would have zero length, making the formula undefined (division by zero).

Point-to-point distance uses the standard 3D distance formula between two coordinates; point-to-plane distance instead measures the shortest perpendicular gap between a point and an entire flat surface.