📐 Distance from Point to Plane Calculator
Calculate shortest distance from a point to a plane
Point P (x₀, y₀, z₀)
Plane Equation: ax + by + cz + d = 0
Example: x + 2y + 3z - 6 = 0 (a=1, b=2, c=3, d=-6)
How to Use This Calculator
Enter Point Coordinates
Input the x, y, and z coordinates of the point from which you want to measure the distance.
Enter Plane Equation
Input the coefficients a, b, c, and d of the plane equation ax + by + cz + d = 0. The normal vector is (a, b, c).
Click Calculate
Press "Calculate Distance" to find the shortest distance from the point to the plane.
Formula
Distance = |ax₀ + by₀ + cz₀ + d| / √(a² + b² + c²)
where (x₀, y₀, z₀) is the point and ax + by + cz + d = 0 is the plane equation
Key Components:
- Point: P(x₀, y₀, z₀)
- Plane: ax + by + cz + d = 0
- Normal vector: n = (a, b, c)
- Normal magnitude: |n| = √(a² + b² + c²)
- Distance is always non-negative (absolute value used)
Example: Point P(1, 2, 3), Plane: x + 2y + 3z - 6 = 0
Distance = |1×1 + 2×2 + 3×3 - 6| / √(1² + 2² + 3²)
= |1 + 4 + 9 - 6| / √(1 + 4 + 9)
= |8| / √14 ≈ 8 / 3.742 ≈ 2.138
About Distance from Point to Plane Calculator
The Distance from Point to Plane Calculator finds the shortest distance from a point in 3D space to a plane. This distance is measured perpendicularly from the point to the plane.
When to Use This Calculator
- Geometry: Calculate distances in 3D space
- Engineering: Measure clearances and gaps in 3D designs
- Physics: Find distances from points to surfaces
- Computer Graphics: Calculate distances for collision detection and rendering
- Mathematics: Solve 3D geometry problems
- Architecture: Measure distances from points to walls or planes
Why Use Our Calculator?
- ✅ Perpendicular Distance: Calculates the shortest (perpendicular) distance
- ✅ Plane Detection: Identifies if the point lies on the plane
- ✅ Step-by-Step Display: Shows the calculation formula with your values
- ✅ Works with All Numbers: Handles decimals, fractions, and negatives
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Completely Free: No registration required
Understanding Distance from Point to Plane
The distance from a point to a plane is the length of the perpendicular segment from the point to the plane. It's always non-negative and equals zero when the point lies on the plane.
- Distance formula: d = |ax₀ + by₀ + cz₀ + d| / √(a² + b² + c²)
- Normal vector: n = (a, b, c) is perpendicular to the plane
- If ax₀ + by₀ + cz₀ + d = 0, the point lies on the plane (distance = 0)
- Distance is measured perpendicularly (shortest path)
- Sign of ax₀ + by₀ + cz₀ + d indicates which side of the plane the point is on
- The absolute value ensures distance is always non-negative
Real-World Applications
Engineering: Calculate clearances between components, minimum distances between objects and walls, and safety margins in 3D designs.
Computer Graphics: Collision detection between objects and surfaces, ray-plane intersection, and distance culling for optimization.
Architecture: Measure distances from points (e.g., fixtures, furniture) to walls, floors, and ceilings in building design.
Frequently Asked Questions
What is the distance from a point to a plane?
The distance is the length of the perpendicular segment from the point to the plane. It's the shortest distance and is always non-negative. Formula: d = |ax₀ + by₀ + cz₀ + d| / √(a² + b² + c²).
What if the point lies on the plane?
If the point satisfies ax + by + cz + d = 0, it lies on the plane and the distance is zero. The calculator will indicate this.
Can the distance be negative?
No! Distance is always non-negative. The formula uses absolute value (|ax₀ + by₀ + cz₀ + d|) to ensure this. However, the sign of ax₀ + by₀ + cz₀ + d indicates which side of the plane the point is on.
What is the normal vector?
The normal vector n = (a, b, c) is perpendicular to the plane. Its magnitude √(a² + b² + c²) is used in the distance formula denominator.
How do I convert a plane from normal form to general form?
If you have a point on the plane and normal vector n = (a, b, c), the plane equation is n · (r - r₀) = 0, which expands to ax + by + cz - (n · r₀) = 0. Rearrange to get d.
What if (a, b, c) = (0, 0, 0)?
This is not a valid plane! The normal vector (a, b, c) cannot be zero. If it's zero, the equation doesn't represent a plane. The calculator will alert you.
How is this different from point-to-line distance?
Point-to-plane distance is for 3D (point and plane). Point-to-line distance is for 2D or 3D (point and line). They use different formulas: plane uses ax+by+cz+d=0, line uses parametric or symmetric equations.