Distance from Point to Plane Calculator

Use the formula d = |Ax0 + By0 + Cz0 + D| / sqrt(A^2 + B^2 + C^2), where Ax + By + Cz + D = 0 is the plane equation and (x0, y0, z0) is the point. For the plane 2x + 3y - z - 6 = 0 and point (1, 2, 3), d = |2(1)+3(2)-1(3)-6| / sqrt(4+9+1) = 1 / 3.742 = 0.267 units.

In the general plane equation Ax + By + Cz + D = 0, the coefficients A, B, and C form the plane's normal vector (a vector perpendicular to the plane), and D is a constant that positions the plane relative to the origin. Any point (x, y, z) satisfying this equation lies on the plane.

Distance is always a non-negative quantity, but Ax0 + By0 + Cz0 + D can be positive or negative depending on which side of the plane the point sits on. Taking the absolute value ensures the formula always returns a positive distance regardless of which side the point is on.

Find two vectors lying in the plane by subtracting one point from each of the other two, then take their cross product to get the normal vector (A, B, C). Substitute one of the three points into Ax + By + Cz + D = 0 to solve for D, giving you the full plane equation to use in the distance formula.

A distance of zero means the point lies exactly on the plane. This occurs whenever the numerator, Ax0 + By0 + Cz0 + D, evaluates to zero, since that term becomes zero regardless of the denominator.

No. The distance between two points uses the Euclidean distance formula, the square root of the sum of squared coordinate differences. This calculator instead finds the shortest distance from a single point to an entire plane (an infinite flat surface), which is always measured along the direction perpendicular to that plane.

Yes, as long as at least one of A, B, or C is nonzero (otherwise the equation does not define a valid plane). If, for example, C = 0, the plane is parallel to the z-axis, and the formula still correctly computes the perpendicular distance using only the nonzero coefficients.

The denominator, the square root of A squared plus B squared plus C squared, is the magnitude (length) of the plane's normal vector (A, B, C). Dividing by this magnitude normalizes the formula so the result is a true distance regardless of how the plane equation happens to be scaled.

A very similar formula applies to the distance from a point to a line in 2D: d = |Ax0 + By0 + C| / sqrt(A^2 + B^2), using the line equation Ax + By + C = 0. It follows exactly the same derivation, just with one fewer dimension.

This calculation is used in 3D computer graphics for collision detection and clipping, in engineering for checking clearances and tolerances against reference surfaces, and in robotics and computer vision for measuring how far an object or sensor reading is from a reference plane.