Distance from Point to Plane Calculator
Enter coordinates of your point (x₀, y₀, z₀) in distance from point to plane calculator. These values represent the position of the point in 3D space.
To calculate the distance from point (2, -1, 3), enter 2 for x₀, -1 for y₀, and 3 for z₀.
Enter coefficients of your plane equation in the form ax + by + cz + d = 0. Enter values for a, b, c, and d. For example, for the plane 2x – 3y + z – 6 = 0, enter 2 for a, -3 for b, 1 for c, and -6 for d.
The calculator displays the distance in units along with your input values.
Distance from Point to Plane Calculator
| Plane Equation | Point Coordinates (x₀, y₀, z₀) | Notes |
|---|---|---|
| Ax + By + Cz + D = 0 | (x₀, y₀, z₀) | Ax₀ + By₀ + Cz₀ + D |
| x = c (vertical plane) | (x₀, y₀, z₀) | x₀ – c |
| y = c (horizontal plane) | (x₀, y₀, z₀) | y₀ – c |
| z = c (horizontal plane) | (x₀, y₀, z₀) | z₀ – c |
Example:
- Plane: 2x + 3y – z + 6 = 0
- Point: (1, -2, 4)
- Distance: D = |2(1) + 3(-2) – 1(4) + 6| / sqrt(2^2 + 3^2 + (-1)^2)
- Calculation: D = |2 – 6 – 4 + 6| / sqrt(4 + 9 + 1) = |2| / sqrt(14).
Point to Plane Distance Formula
|ax₀ + by₀ + cz₀ + d|/√(a² + b² + c²)Parameters
- (x₀, y₀, z₀) = Coordinates of the point
- a, b, c = Coefficients of the plane equation
- d = Constant term in the plane equation
- √(a² + b² + c²) = Magnitude of the normal vector to the plane
For point (1, 2, -1) and plane 3x – 2y + z + 4 = 0:
Distance = |3(1) - 2(2) + (-1) + 4|/√(3² + (-2)² + 1²) = |3 - 4 - 1 + 4|/√(14) = |2|/√14 ≈ 0.5345 unitsHow to Calculate Distance from Point to Plane?
- Write the plane equation in standard form (ax + by + cz + d = 0)
- Identify the point coordinates (x₀, y₀, z₀)
- Calculate the numerator: |ax₀ + by₀ + cz₀ + d|
- Calculate the denominator: √(a² + b² + c²)
- Divide the numerator by the denominator
For point (0, 1, 2) and plane x + y + z = 3: First convert plane to standard form: x + y + z – 3 = 0 Then apply formula: |1(0) + 1(1) + 1(2) – 3|/√(1² + 1² + 1²) = |0 + 1 + 2 – 3|/√3 = 0/√3 = 0 units
What is Distance from Point to Plane?
The distance from a point to a plane is the shortest possible distance between any point and a plane in three-dimensional space. It represents the length of the perpendicular line segment drawn from the point to the plane. This distance is always the shortest possible path from the point to any point on the plane.
A point in 3D space is represented by its three coordinates (x, y, z). A plane is a flat, two-dimensional surface that extends infinitely in three-dimensional space and is typically represented by an equation of the form ax + by + cz + d = 0, where a, b, and c cannot all be zero.