Distance from Point to Plane Calculator

This distance from point to plane calculator computes the shortest distance from a point (x₁, y₁, z₁) to a plane the equation Ax + By + Cz + D = 0, using formula: Distance = |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²).

Consider a plane defined by the equation:

3x + 2y - z + 4 = 0

And a point P(2, -1, 3) in space.

Distance from Point to Plane Calculator

Plane Equation	Point Coordinates	Distance (units)
x + y + z = 1	P(0, 0, 0)	0.577
2x – y + z = 4	P(1, 1, 1)	1.633
3x + 2y – z = 6	P(2, -1, 3)	2.408
x – 2y + 2z = 3	P(0, 2, 1)	1.225
4x + 4y + 4z = 12	P(1, 1, 1)	0.000

For example,

Distance from Point to Plane Formula

The formula for calculating the distance from a point to a plane is:

|Ax₀ + By₀ + Cz₀ + D|
d = ───────────────────────
    √(A² + B² + C²)

Where:

• (x₀, y₀, z₀) are the coordinates of the point
• Ax + By + Cz + D = 0 is the equation of the plane
• A, B, and C are the coefficients of the plane equation
• D is the constant term
• |…| represents the absolute value
• √ represents the square root

Given a plane 2x – 3y + 4z + 6 = 0 and point P(1, 2, -1):

d = |2(1) - 3(2) + 4(-1) + 6|
    ─────────────────────────
    √(2² + (-3)² + 4²)

d = |2 - 6 - 4 + 6|
    ───────────────
    √(4 + 9 + 16)

d = |-2|
    ─────
    √29

d = 2/√29 ≈ 0.371 units

How to find Distance from Point to Plane?

Following these detailed steps:

• Identify the plane equation in standard form (Ax + By + Cz + D = 0)
• Note the point coordinates (x₀, y₀, z₀)
• Extract the coefficients A, B, C, and D
• Substitute values into the formula
• Calculate the numerator and denominator separately
• Perform the final division

Given plane: x + 2y – 2z – 4 = 0 and point P(3, 0, 1):

• Identify coefficients:
  • A = 1
  • B = 2
  • C = -2
  • D = -4
• Calculate numerator: |1(3) + 2(0) – 2(1) – 4| |3 + 0 – 2 – 4| |-3| = 3
• Calculate denominator: √(1² + 2² + (-2)²) √(1 + 4 + 4) √9 = 3
• Final calculation: d = 3/3 = 1 unit

Sources and References

• MIT OpenCourseWare – Linear Algebra: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/
• Paul’s Online Math Notes – Distance to Planes: https://tutorial.math.lamar.edu/

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