Point to Plane Distance Calculator
Calculate the shortest (perpendicular) distance from a 3D point to a plane given in general form Ax + By + Cz + D = 0, plus the foot of the perpendicular — the closest point on the plane.
Input
Plane equation: Ax + By + Cz + D = 0.
Output
| Metric | Value |
|---|---|
| No data yet | |
More ways to use this tool
REST API
curl -X POST https://api.iotools.cloud/v1/tool/point-to-plane-distance-calculator \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"a": "2",
"b": "-1",
"c": "2",
"d": "-3",
"x0": "1",
"y0": "2",
"z0": "3"
}'Swap in your own key from your account. The tool's fields are the body — no wrapper.
Ask an AI agent
Use the IOTools `point-to-plane-distance-calculator` tool (Point to Plane Distance Calculator) on this input:
YOUR_INPUT_HEREPaste this at any agent connected to the IOTools MCP server, then add your input.
Embed widget
<iframe
src="https://iotools.cloud/embed/point-to-plane-distance-calculator/"
width="100%" height="520" frameborder="0" scrolling="no" loading="lazy"
title="Point to Plane Distance Calculator — iotools.cloud"
sandbox="allow-scripts allow-forms allow-same-origin allow-downloads allow-popups allow-popups-to-escape-sandbox"
allow="clipboard-write"
style="width:100%;border:1px solid #e5e7eb;border-radius:12px;overflow:hidden"></iframe>
<script src="https://iotools.cloud/embed.js" async></script>Drop this into your own page — free, no key required, just a link back.
| Cost per API/MCP call | From 5 credits |
|---|---|
| Need more credits? | View pricing |
Also available with
Guides
The Point to Plane Distance Calculator finds the shortest (perpendicular) distance from a point in 3D space to a flat plane, plus the exact foot of the perpendicular — the closest point on the plane to your point. It's the tool you reach for when checking clearance from a surface, projecting a point onto a plane, or verifying a plane equation in coursework or CAD/engineering work.
How to use it
- Enter the plane's coefficients A, B, C, and D from its general-form equation
Ax + By + Cz + D = 0. - Enter the point's coordinates x₀, y₀, z₀.
- Read the results — the shortest distance, the foot of the perpendicular, and the signed distance — updated automatically as you type.
A, B, and C can't all be zero (that wouldn't describe a plane). The point can be on either side of the plane, or on it.
The formula
For a plane Ax + By + Cz + D = 0 and a point (x₀, y₀, z₀), the perpendicular distance is:
distance = |A·x₀ + B·y₀ + C·z₀ + D| / √(A² + B² + C²)
The numerator measures how far the point is from satisfying the plane equation; dividing by the magnitude of the plane's normal vector (A, B, C) converts that into an actual spatial distance. Dropping the absolute value gives the signed distance — positive on one side of the plane, negative on the other, zero exactly on it.
The foot of the perpendicular — the point on the plane closest to (x₀, y₀, z₀) — is found by moving from the point along the plane's normal direction by the signed distance:
foot = (x₀, y₀, z₀) − t·(A, B, C), where t = (A·x₀ + B·y₀ + C·z₀ + D) / (A² + B² + C²)
Worked example
For the plane 2x − y + 2z − 3 = 0 and the point (1, 2, 3): the numerator is 2(1) − 1(2) + 2(3) − 3 = 3, and the normal's magnitude is √(2² + (−1)² + 2²) = √9 = 3. So the distance is |3| / 3 = 1. The foot of the perpendicular works out to (0.333333, 2.33333, 2.33333).
Frequently asked questions
What if my plane equation isn't in Ax + By + Cz + D = 0 form?
Rearrange it first. A plane through three points, or given in a different form (like point-normal form), can always be converted to the general form before entering the coefficients here.
Why does the calculator also give a "signed distance"?
The sign tells you which side of the plane the point falls on relative to the normal vector (A, B, C) — useful when you need to know not just "how far" but "which side," such as checking whether a point is above or below a reference plane.
Can the point lie on the plane?
Yes — if it does, both the distance and signed distance come out to 0, and the foot of the perpendicular equals the point itself.
Is my data sent anywhere?
No. All calculations run locally in your browser. Your coordinates never leave your device.