The point to plane distance calculator finds the shortest distance from a point in space to a flat surface, which is always measured along the perpendicular. Any other path from the point to the plane is longer, and the perpendicular is the only one whose length can be written in a single closed formula.
Beyond the distance itself, this tool returns the signed distance — which tells you which side of the plane the point sits on — and the foot of the perpendicular, the exact coordinates of the closest point on the plane. Arb Digital built it with both because in practice the foot is usually what you actually need: it is the projection, the contact point, the shadow, the snapped position.
What This Point to Plane Distance Calculator Does
Enter a point and a plane. The plane can be given directly as the four coefficients of ax + by + cz + d = 0, or as three points lying on it, in which case the tool derives the coefficients for you. It then returns the unsigned distance as the headline figure, the signed distance, the side of the plane the point falls on, the raw substitution value, the length of the normal vector, the foot of the perpendicular and the mirror image of the point reflected through the plane.
The sign convention is stated explicitly on the page because it is where most errors start. This tool uses ax + by + cz + d = 0, so a plane written in the equally common form ax + by + cz = k has d = −k. Enter 2x − 2y + z = 5 as a = 2, b = −2, c = 1, d = −5, and the hint under the coefficient row says so.
Degenerate input is handled in words. If a, b and c are all zero, there is no plane at all — the equation reduces either to a contradiction or to a statement about every point in space — and the tool says which rather than dividing by zero. In three-point mode, collinear points produce the same message for the same underlying reason.
How to Use It
- Enter the point's coordinates. These are the x, y and z of the point you are measuring from.
- Choose how the plane is specified — four coefficients, or three points lying on the surface.
- Fill in the coefficients, watching the sign of d, or paste the three points one per line as x,y,z.
- Read the distance as the headline figure and check the sign in the grid if you need to know which side of the plane you are on.
- Take the foot of the perpendicular from the detail panel if you need the closest point on the plane rather than just the distance to it.
The Formula: How It Is Calculated
For a plane ax + by + cz + d = 0 and a point (x₀, y₀, z₀), the perpendicular distance is
D = |a·x₀ + b·y₀ + c·z₀ + d| ÷ √(a² + b² + c²),
which MathWorld gives in the same form. The numerator is what you get by substituting the point into the left-hand side of the equation, and the denominator is the length of the normal vector (a, b, c). Dropping the absolute value gives the signed distance, whose sign tells you whether the point lies on the side the normal points toward or the opposite side.
The derivation is short. Take any point Q on the plane. The displacement from Q to your point P is P − Q, and the distance you want is the component of that displacement along the unit normal — that is, the dot product of (P − Q) with n̂. Expanding that dot product and using the fact that Q satisfies the plane equation makes every Q-dependent term collapse into d, leaving the formula above. The answer does not depend on which Q you chose, which is the geometric content of the result.
The Foot of the Perpendicular, and Why It Matters More Than the Distance
The foot of the perpendicular is the point on the plane closest to your point, and it is found by stepping from P back along the normal by exactly the signed distance: F = P − (s ÷ |n|²)·n, where s is the substitution value a·x₀ + b·y₀ + c·z₀ + d. It is the orthogonal projection of P onto the plane. The plane itself, and the Hessian normal form that makes this projection immediate, are set out in MathWorld’s article on the plane.
That projection is what most real applications are after. In graphics it is where a shadow lands under a directional light perpendicular to the surface. In CAD it is where a point snaps onto a face. In machine learning it is the projection onto the decision boundary of a linear classifier, which is exactly how the margin of a support vector machine is defined. In surveying it is the reduction of an observation to a reference plane. The distance is a byproduct of the projection at least as often as the projection is a byproduct of the distance.
The tool also reports the reflection of the point through the plane, which is the foot stepped through by the same amount again: P′ = P − 2(s ÷ |n|²)·n. Reflections come up in mirror geometry, in ray tracing, and in any symmetry argument where you need the image of a point across a surface.
What the Sign Actually Tells You
The signed distance is positive when the point lies on the side that the normal vector (a, b, c) points toward, and negative on the other side. That makes it a half-space test as well as a measurement, which is why it appears everywhere in collision detection, view-frustum culling and constructive solid geometry: a single scalar tells you both how far and which way.
The trap is that the sign depends on the arbitrary choice of normal direction. Multiply the whole equation by −1 and you have the same plane, the same unsigned distance, and the opposite sign on everything. If you are testing which side of a plane a point is on, you must fix and document the normal's direction first, or the answer is meaningless. Our plane equation calculator reports the normal it derived for exactly this reason.
A signed distance of zero means the point lies on the plane. In floating-point arithmetic that will rarely come out as exactly zero for computed input, so real code compares against a small tolerance rather than testing for equality — and choosing that tolerance sensibly means scaling it to the size of the coordinates involved, not picking a fixed number and hoping.
Where This Sits Among the Other 3D Tools
This page measures from a point to a plane you already have. If you need to build the plane first, from three points or from a point and a normal, that is the plane equation calculator, which returns the coefficients in general, vector, Hessian and parametric form. The two pages are designed to be used in sequence: build there, measure here.
For the straight-line separation between two points in space with no plane involved, use the 3D distance calculator. The cross product calculator produces the normal from two edge vectors on its own, and the dot product calculator handles the projection step that sits at the heart of the derivation above. General vector arithmetic is in the vector calculator, and coordinate conversion comes before all of it if your data starts out in cylindrical or spherical form.
Related Distance Problems That Look Similar and Are Not
Distance from a point to a line in three dimensions is a different formula: it uses the magnitude of a cross product rather than a dot product, because the perpendicular from a point to a line is not determined by a single fixed direction. Distance between two skew lines uses the scalar triple product. Distance between two parallel planes is the difference of their constants after both have been scaled to a common unit normal — which reduces to this page's formula applied to any point on one of them.
Distance from a point to a line segment or to a bounded polygon is different again, and it is where naive implementations most often go wrong. The perpendicular foot may fall outside the segment or outside the polygon's boundary, in which case the closest point is an endpoint or an edge, and the perpendicular distance is not the answer at all. This tool treats the plane as infinite, which is the standard mathematical object; if your surface is a bounded triangle or quad, check whether the foot lands inside it before using the distance.
Arb Digital builds browser-native tools that run without frameworks or server calls. If your engineering, education or SaaS site needs working geometry rather than a static formula page, we can build it.
Web Design Services Talk to Arb DigitalCommon Mistakes to Avoid
- Getting the sign of d wrong — the form here is ax + by + cz + d = 0, so a plane written as 2x − 2y + z = 5 has d = −5, not +5.
- Forgetting to divide by the normal's length — the raw substitution value is proportional to the distance but equals it only when the normal is a unit vector.
- Treating the sign as absolute — it depends entirely on the direction chosen for the normal, and negating the whole equation flips it without changing the plane.
- Using this formula on a bounded surface — for a triangle or a polygon the perpendicular foot may land outside the boundary, in which case the nearest point is on an edge instead.
- Confusing point-to-plane with point-to-line — the line version uses a cross product magnitude, and substituting one formula for the other produces a plausible wrong number.
Related Free Tools From Arb Digital
Build the plane first with the plane equation calculator, get the normal alone from the cross product calculator, handle projections with the dot product calculator, measure straight-line separation with the 3D distance calculator, do general vector work in the vector calculator, and convert between coordinate systems with the coordinates converter. Everything else is in the free online tools hub.
Frequently Asked Questions
For a plane ax + by + cz + d = 0 and a point (x₀, y₀, z₀), the distance is the absolute value of a·x₀ + b·y₀ + c·z₀ + d divided by the square root of a² + b² + c². The numerator substitutes the point into the equation; the denominator is the length of the normal vector.
Positive means the point lies on the side the normal vector (a, b, c) points toward, and negative means the opposite side. Because negating the whole equation gives the same plane with the opposite normal, the sign is only meaningful once the normal's direction has been fixed.
Step from your point back along the normal by the signed distance. The foot of the perpendicular is F = P − (s ÷ |n|²)·n, where s is the value of a·x₀ + b·y₀ + c·z₀ + d. That point is the orthogonal projection of P onto the plane.
Because this calculator uses the form ax + by + cz + d = 0, so that plane has d = −k. Writing 2x − 2y + z = 5 requires entering d as −5. Getting this sign wrong shifts the plane and changes the distance.
Yes. Any other path from the point to the plane forms the hypotenuse of a right triangle whose other leg lies in the plane, so it is strictly longer. This is why the perpendicular distance is the only one that has a single closed-form expression.
Not directly. The formula treats the plane as infinite in every direction. If your surface is a bounded triangle or polygon, check whether the foot of the perpendicular lands inside the boundary; if it does not, the closest point is on an edge or at a vertex instead.
There is no plane. The equation collapses to d = 0, which is either always true, describing all of space, or never true, describing nothing. The tool reports this in words rather than dividing by a zero normal length.
This page performs exact geometric algebra on the values you enter and treats the plane as infinite in extent. Results are computed in double-precision floating point, so a point intended to lie exactly on the plane will usually return a very small non-zero distance rather than zero.
