Distance from Point to Plane Calculator (3D)

Calculate the shortest perpendicular distance for a plane in standard form, point-normal form, or from three points, with steps and the closest point shown.

Enter the point and plane

Use real numbers and a period for decimals. Negative values may be entered directly, including a negative D constant.

Try an example
Point coordinates
Plane representation

Enter Ax + By + Cz + D = 0; for example, 2x − y + 2z − 3 = 0.

Standard-form plane coefficients
Exact form is also shown when practical.
Enter all required values or choose an example to see the complete solution.

Labeled geometry preview

Schematic view — not drawn to scale.
Your browser does not support the canvas diagram. The textual geometry summary appears below it.

Enter valid values to generate a textual summary of the plotted point, plane, closest point, normal, and distance.

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How to use the calculator

  1. Choose the plane representation. For standard form, read A, B, C, and D from Ax + By + Cz + D = 0.
  2. Enter the point and plane values. Include the minus sign when a coordinate or constant such as D is negative.
  3. Select the decimal precision and optionally emphasize signed distance.
  4. Press Calculate Distance, or pause after completing every required field for automatic calculation.
  5. Interpret the unsigned distance as “how far,” the sign as the oriented side, and the perpendicular foot as the closest point on the plane.

Learn more: distance from a point to a plane (3D)

In analytic geometry, the shortest distance from a point to a plane is measured along a line that is perpendicular to the plane. If the plane is written in standard form A x + B y + C z + D = 0 and the point is P = (x₀, y₀, z₀), the (unsigned) distance is 

d = |A x₀ + B y₀ + C z₀ + D| / √(A² + B² + C²)

This formula is scale-invariant: multiplying all plane coefficients by the same nonzero number doesn’t change d, because numerator and denominator scale together. The sign of A x₀ + B y₀ + C z₀ + D tells you on which side of the oriented plane the point lies; drop the absolute value to get the signed distance: 

dₛ = (A x₀ + B y₀ + C z₀ + D) / √(A² + B² + C²)

Where does the formula come from?

The vector n = (A, B, C) is a normal to the plane. Any point Q on the plane satisfies A Qₓ + B Qᵧ + C Q_z + D = 0. The displacement from Q to P is v = P − Q. Projecting v onto n gives the component perpendicular to the plane: (v · n)/‖n‖. Replace v · n with (P · n) − (Q · n) = A x₀ + B y₀ + C z₀ + D and take absolute value to get the shortest (unsigned) distance. The same projection also yields the perpendicular foot (closest point) on the plane: 

k = (A x₀ + B y₀ + C z₀ + D) / (A² + B² + C²)
F = (x₀ − A k,  y₀ − B k,  z₀ − C k)

Other plane descriptions you can use

  • Point + normal: given a plane point Pₚ and normal n, set (A,B,C) = n and D = −n · Pₚ, then apply the core formula.
  • Three points: with non-collinear P₁, P₂, P₃, compute n = (P₂ − P₁) × (P₃ − P₁), then D = −n · P₁. Collinear or repeated points are invalid.

Worked example

Let P = (2, −1, 0.5) and the plane 2x − y + 2z − 3 = 0. Then n = (2, −1, 2), ‖n‖ = √(4 + 1 + 4) = 3, and the numerator is |2·2 + (−1)·(−1) + 2·0.5 − 3| = |4 + 1 + 1 − 3| = 3. Therefore d = 3 / 3 = 1. The foot uses k = (A x₀ + B y₀ + C z₀ + D)/(A²+B²+C²) = 3/9 = 1/3, so F = (2 − 2·1/3, −1 − (−1)·1/3, 0.5 − 2·1/3) = (4/3, −2/3, −1/6).

Numerical tips & pitfalls

  • Degenerate planes: standard form with A=B=C=0 or a zero normal vector is invalid.
  • Stability: if your coordinates are very large or very small, distances can lose precision in floating-point; rescale your data (meters → kilometers, etc.) if needed.
  • Normalization: you do not need to normalize (A,B,C) because the formula divides by ‖n‖ automatically.
  • Signed vs. unsigned: signed distance is handy for orientation tests (e.g., which side of a cutting plane a point lies on); for pure “how far,” use the unsigned value.

Where this shows up

Point-to-plane distance is everywhere: CAD “snap to surface,” camera pose residuals in computer vision, collision detection and steering behaviors in robotics/games, fitting planes to noisy 3D scans, geospatial offsets to terrain surfaces, and quality checks in manufacturing metrology.

Privacy note: everything on this page runs locally in your browser—no inputs are uploaded.

Common cases and interpretations

Useful simplified formulas

  • Coordinate planes: d(xy)=|z₀|, d(xz)=|y₀|, and d(yz)=|x₀|.
  • Point on the plane: if Ax₀+By₀+Cz₀+D=0, then d=0.
  • Plane through the origin (D=0): d=|Ax₀+By₀+Cz₀|/‖n‖.
  • Point-normal form: for plane point Q and normal n, d=|(P−Q)·n|/‖n‖.
How to read the output
OutputInterpretation
d = 0The point lies on the plane; the foot equals the point.
Positive signed distanceThe point is on the side toward the chosen normal.
Negative signed distanceThe point is on the side opposite the chosen normal.
UnitsThe distance preserves the coordinate units when all inputs use the same unit.

Frequently asked questions

What formula does the calculator use?

For a plane Ax + By + Cz + D = 0 and point (x0, y0, z0), the perpendicular distance is |Ax0 + By0 + Cz0 + D| / sqrt(A^2 + B^2 + C^2).

Can I enter the plane in different forms?

Yes. Enter Ax + By + Cz + D = 0, a point and normal vector, or three non-collinear points. The calculator converts the latter two forms to standard form.

How do I calculate point-to-plane distance by hand?

Substitute the point coordinates into Ax + By + Cz + D, take the absolute value, calculate the normal magnitude sqrt(A^2 + B^2 + C^2), and divide the two results.

What is the distance to the xy-, xz-, or yz-plane?

For point (x0, y0, z0), the distances to the xy-, xz-, and yz-planes are |z0|, |y0|, and |x0| respectively.

What is the distance from the origin to a plane?

For Ax + By + Cz + D = 0, the distance from (0, 0, 0) is |D| / sqrt(A^2 + B^2 + C^2).

What does a distance of zero mean?

A zero distance means the point satisfies the plane equation and therefore lies on the plane. The perpendicular foot is the point itself.

What units does the result use?

The distance and perpendicular-foot coordinates use the same linear units as the input coordinates, provided all coordinates use one consistent unit.

What is signed distance?

Signed distance keeps the sign of Ax0 + By0 + Cz0 + D relative to the chosen normal (A, B, C). Its magnitude is the ordinary distance; reversing the normal reverses the sign.

What is the foot of the perpendicular?

It is the closest point on the plane to the submitted point. The segment from the point to this foot is parallel to the plane normal.

What happens if the three plane points are collinear?

Collinear or repeated points do not define a unique plane, so the calculator reports an error and asks for three non-collinear points.

Is my data private?

Everything runs entirely in your browser. No input is uploaded.

Methodology and editorial information

Content owner: Starlight Tools, a Starlight Robotics project publishing browser-based calculation and utility tools. Last reviewed: 17 July 2026. No named individual or independent reviewer is currently listed.

The calculator uses IEEE 754 browser arithmetic and accepts finite real-number inputs. Verification tolerances scale with the submitted values; a displayed “pass” means the residual is within floating-point tolerance, not that arithmetic is symbolic or error-free. Exact forms are produced for practical integer-input cases, while decimal approximations use the selected precision.

References: OpenStax, Calculus Volume 3: Equations of Lines and Planes in Space; MIT OpenCourseWare, Distances to Planes and Lines.

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