Point to plane distance calculator
The point (1, 1, 1) sits four thirds of a unit, 1.333333333333, from the plane 2x + y + 2z − 9 = 0. Enter the plane's four coefficients and a point in space to get the shortest gap, its sign and an on-plane check. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- A
- 2
- B
- 1
- C
- 2
- D
- -9
- X
- 1
- Y
- 1
- Distance
- 1.3333333333
- Signed distance
- -1.3333333333
- Norm
- 3
The perpendicular distance from a point to a plane is the shortest path from that point to any location on the infinite plane. You need it when a student checks homework on the general plane equation, a surveyor or CAD operator measures clearance from a floor or wall plane, a game or robotics pipeline keeps an actor clear of a planar constraint, or a machine-vision stack reports offset from a fitted surface. This capability accepts the seven numbers a, b, c, d, x, y, and z that define the plane ax plus by plus cz plus d equals zero and the query point, rejects the degenerate case where the plane normal (a, b, c) is the zero vector, and returns the non-negative distance together with the signed distance, the normal length, and whether the point already lies on the plane.
What the formula is and how the numbers work
Write the infinite plane in general (Cartesian) form as a times x plus b times y plus c times z plus d equals zero. The triple (a, b, c) is a normal vector to the plane: it points perpendicular to every direction that stays on the plane. The length of that normal is the Euclidean norm sqrt of a squared plus b squared plus c squared.
For a query point (x, y, z), plug the coordinates into the left-hand side to get the residual a times x plus b times y plus c times z plus d. The absolute value of that residual divided by the normal length is exactly the perpendicular distance from the point to the plane. When the residual is zero, the point satisfies the plane equation and the distance is zero.
A classic sanity check is the plane two x plus y plus two z minus nine equals zero (a 2-1-2 normal of length three) and the point (1, 1, 1): residual absolute value four, norm three, distance four thirds. Every step is elementary arithmetic and one three-dimensional square root (or hypot), so there is no iteration, no sampling, no dependence on wall-clock time, and no need for point-normal conversion inside the input contract once you already hold a, b, c, and d.
What you can do with it
Clearance and setback from a floor, wall, or roof plane
Measure how far a structure, sensor reading, or mesh vertex sits from a construction or survey plane expressed in general form.
Game, CAD, and robotics offset from a planar constraint
Compute lateral offset of a character, part, or end-effector from an infinite plane used as a collision, floor, or guidance constraint.
FAQ
What is signed distance to a plane?
Signed distance is (ax+by+cz+d)/sqrt(a²+b²+c²). Its sign tells which half-space the point is in relative to the oriented normal (a,b,c); magnitude matches the ordinary distance.
Does this work for a triangle or bounded face?
No. This capability measures distance to the infinite plane. Shortest distance to a finite polygon or mesh face needs clamp logic and is a different problem.
Is the point to plane distance calculator free?
The browser tool is free and runs locally.
What if the first three coefficients are all zero?
Then no plane exists: the expression shrinks to the constant term alone. The calculator refuses to produce a distance in that case, so make sure at least one of the first three coefficients is non-zero.
What is the normal of the plane and why does its length matter?
The first three coefficients form a vector standing at right angles to the plane. The distance is the plugged-in value divided by that vector's length, which is 3 for the plane 2x + y + 2z − 9 = 0.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/geom/point-plane-distance \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":2,"b":1,"c":2,"d":-9,"x":1,"y":1,"z":1}'const res = await fetch("https://api.kit.forhosting.com/geom/point-plane-distance", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 2,
"b": 1,
"c": 2,
"d": -9,
"x": 1,
"y": 1,
"z": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/point-plane-distance",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 2,
"b": 1,
"c": 2,
"d": -9,
"x": 1,
"y": 1,
"z": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/point-plane-distance", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":2,"b":1,"c":2,"d":-9,"x":1,"y":1,"z":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":2,"b":1,"c":2,"d":-9,"x":1,"y":1,"z":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/point-plane-distance", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a": 2,
"b": 1,
"c": 2,
"d": -9,
"x": 1,
"y": 1,
"z": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.point_plane_distance",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Limits
max_abs | 1000000000000 |
min_norm | 1e-15 |
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Results are deterministic JSON—no computer-algebra system, no random seeds, and the same pure solve function in the free browser tool and the prepaid API path—so homework checks, grading scripts, and production pipelines all see the same numbers every time.
What the formula is and how the numbers work
When a, b, and c are all zero, the expression collapses to a constant d and does not define a plane at all—this tool returns invalid_input instead of inventing a distance. Another check is any point known to lie on the plane: distance must be zero and on_plane true.
How to call the API and what comes back
Use it from solid-geometry homework pipelines, construction clearance scripts, game-level tooling, interview-prep sandboxes, robotics keep-out checks, or any workflow that needs a reproducible point-to-plane distance without installing a computer-algebra package on every machine. Send a JSON body with seven numeric fields. Field a is the first coefficient of the plane normal; aliases A and coeff_a are accepted. Field b is the second coefficient; aliases B and coeff_b map to the same meaning. Field c is the third coefficient; aliases C and coeff_c are accepted. Field d is the constant term; aliases D and coeff_d work. Field x is the point's first coordinate; aliases x0, px, and point_x work. Field y is the second coordinate; aliases y0, py, and point_y work. Field z is the third coordinate; aliases z0, pz, and point_z work. On success the response echoes a, b, c, d, x, y, and z, adds distance (non-negative), signed_distance (positive or negative according to which side of the oriented normal the point sits on), on_plane (true when distance is zero), norm (sqrt of a squared plus b squared plus c squared), method set to perpendicular, and a short formula block documenting the closed forms. Failed validation—missing a required field, a non-finite number, a value outside published absolute limits, or a normal (a, b, c) that is entirely zero—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item and zero marginal unit price.
Practical tips, limits, and related searches
If your plane is given by three non-collinear points, form two edge vectors, take their cross product to obtain (a, b, c), then set d so that one of the three points satisfies the equation. When teaching, prefer paired examples: two-one-two-minus-nine at (1,1,1) for distance four thirds; a point on the plane for distance zero; and a equals b equals c equals zero to demonstrate the hard rejection path that protects callers from a non-plane. Prefer this capability when you already have the general form of an infinite plane in three dimensions. Keep coefficients and coordinates inside the published absolute range up to one trillion so intermediate products stay finite in IEEE double precision on the Worker and in the browser. Units are whatever you pass—metres, feet, or game units—as long as a, b, c, d, x, y, and z are consistent with a single length unit for the distance result; the API does not convert units for you. Because the formula only needs seven scalars, students can recompute the residual and the hypot by hand for small integers and match the API line by line. This landing targets people searching for a point to plane distance calculator or perpendicular distance to ax plus by plus cz plus d equals zero; it is not a distance-to-triangle or distance-to-half-space clipping tool—shortest distance to a finite polygonal patch needs barycentric clamp logic that is out of scope here. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when users type a vanishing normal while learning the contract. Round-trip checks are easy: after solving, confirm that distance equals the absolute value of signed_distance, that on_plane is true exactly when distance is zero, and that norm squared equals a squared plus b squared plus c squared within floating tolerance.
Integration use cases
Homework and exam checks for point-plane distance
Give students a plane ax+by+cz+d=0 and a point; compare their hand-computed perpendicular distance against a deterministic API or free browser result.
Technical questions
What is the point to plane distance formula?
For the plane ax+by+cz+d=0 and point (x,y,z), the perpendicular distance is |ax+by+cz+d| / sqrt(a²+b²+c²). The absolute value ensures a non-negative length.
Why is a zero plane normal rejected?
If a=b=c=0 the equation reduces to d=0 or a contradiction and does not define a plane. This tool returns invalid_input instead of a meaningless distance.