Enter a point and plane details for instant accurate geometry results. Switch input methods easily. View distance, foot point, and worked geometry values clearly.
| Case | Point | Plane | Method | Distance |
|---|---|---|---|---|
| Example 1 | (2, -1, 4) | 3x - 2y + 6z - 12 = 0 | Coefficients | 2.8571 |
| Example 2 | (3, 1, 2) | x + 2y - z - 4 = 0 | Coefficients | 0.4082 |
| Example 3 | (2, 1, 3) | x + y + z - 1 = 0 | From three plane points | 2.8868 |
For Example 3, the plane can come from points (0,0,1), (1,0,0), and (0,1,0).
For a plane written as Ax + By + Cz + D = 0 and a point (x₀, y₀, z₀), the point to plane distance is:
Distance = |Ax₀ + By₀ + Cz₀ + D| / √(A² + B² + C²)
The signed distance keeps the sign of the numerator. The absolute distance removes direction and keeps only magnitude.
The foot of the perpendicular is found by moving from the point along the plane normal:
Foot = Point - ((Ax₀ + By₀ + Cz₀ + D) / (A² + B² + C²)) × (A, B, C)
If you define the plane with three points, the normal vector is created by the cross product of two plane direction vectors.
A distance point plane calculator finds the shortest gap between a point and a plane in three-dimensional space. That gap is always perpendicular. This makes the result dependable for geometry, graphics, physics, mapping, and technical study. Instead of measuring along a random line, the calculator uses the plane normal. That gives a true geometric distance.
You can enter the plane in two practical ways. The first method uses plane coefficients in the standard form Ax + By + Cz + D = 0. The second method builds the plane from three known points. Both options help students verify homework, compare methods, and understand how coordinate geometry connects with vector operations.
The tool returns more than one number. It shows the absolute distance, the signed distance, the plane equation, the numerator, the denominator, and the foot of the perpendicular. These outputs explain direction as well as size. A signed result reveals which side of the plane the point occupies. The foot point shows the nearest location directly on the plane.
This calculator is useful in analytic geometry classes. It also helps with 3D modeling, collision checks, spatial analysis, and coordinate validation. When you project a point onto a plane, you often need the exact nearest point. The displayed foot point handles that task and makes manual checking much easier during practice or review.
Because the form supports custom decimal precision, you can control rounding for classroom work or technical review. The small unit field lets you label results with meters, feet, millimeters, or any chosen unit. That keeps the output readable when your coordinate system represents real measurements rather than abstract values.
If you define a plane from three points, the calculator first creates two direction vectors. It then takes their cross product. That cross product becomes the normal vector of the plane. After that step, the same distance formula applies. This keeps both input modes mathematically consistent and easy to compare.
Use this page when you need a quick and transparent point to plane distance solution. The formula is standard, the steps are visible, and the results are easy to export. With the worked outputs, example table, and concise explanations, the calculator helps you learn the method and apply it with confidence.
It uses the standard 3D point to plane distance formula: |Ax₀ + By₀ + Cz₀ + D| divided by √(A² + B² + C²).
Signed distance keeps the positive or negative sign of the numerator. It shows which side of the plane the point lies on.
Yes. Negative values are valid for point coordinates, plane coefficients, and the three plane points used to define a plane.
If A, B, and C are all zero, the equation has no plane normal. Without a normal vector, distance cannot be computed.
This is useful when your geometry problem provides coordinates instead of coefficients. The calculator converts those points into a valid plane equation.
It is the closest point on the plane to the given point. It lies on the normal line dropped from the point.
No. The unit label is only for display. It helps interpret the result when coordinates represent real measurements.
Use lower precision for quick checks and higher precision for assignments, reports, or cases where rounding differences matter.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.