Distance from a Point to a Plane (3D) Calculator

This Distance from a Point to a Plane (3D) Calculator helps you find the shortest distance from a point to a plane in three-dimensional space. Substitute the point coordinates and plane coefficients into the distance formula, then simplify the expression to get the final distance. This measures the perpendicular distance from the point to the plane. It is a simple way to check answers, understand the method clearly, and practise 3D coordinate geometry step by step.

Step-by-step method

1. Identify the point P(x₀,y₀,z₀) and plane coefficients A,B,C,d.
2. Substitute into the distance formula.
3. Simplify to get the distance.

Formula:

Example 1: (1,2,3); 2x-3y+4z=10

Step 1 - Identify the point and plane coefficients.

In this problem: Read x₀,y₀,z₀ from the point and A,B,C,d from Ax+By+Cz=d.

Step 2 - Write the distance formula.

In this problem: Use Distance = |Ax₀ + By₀ + Cz₀ − d| / √(A² + B² + C²).

Step 3a - Substitute values.

In this problem: Substitute A,B,C,d and the point coordinates into the formula.

Step 3b - Simplify inside absolute value and square root.

In this problem: Compute the numerator and the squared-sum denominator.

Step 3c - Simplify to the final distance.

In this problem: Simplify |…| and √(…).

Final answer: Distance = 2*sqrt(29)/29

Example 2: (0,0,0); x+y+z=6

Step 1 - Identify the point and plane coefficients.

In this problem: Read x₀,y₀,z₀ from the point and A,B,C,d from Ax+By+Cz=d.

Step 2 - Write the distance formula.

In this problem: Use Distance = |Ax₀ + By₀ + Cz₀ − d| / √(A² + B² + C²).

Step 3a - Substitute values.

In this problem: Substitute A,B,C,d and the point coordinates into the formula.

Step 3b - Simplify inside absolute value and square root.

In this problem: Compute the numerator and the squared-sum denominator.

Step 3c - Simplify to the final distance.

In this problem: Simplify |…| and √(…).

Final answer: Distance = 2*sqrt(3)