Distance from Point to Plane Calculator
Calculate the perpendicular distance from a point to a plane in 3D space
Point and Plane Input
Point Coordinates (a, b, c)
How is your plane given?
Plane Equation: Ax + By + Cz + D = 0
Note: A, B, and C cannot all be zero for a valid plane equation.
Distance Result
Point: (0, 0, 0)
Plane: 1x + 1y + 1z + 0 = 0
Formula: d = |Aa + Bb + Cc + D| / √(A² + B² + C²)
Step-by-Step Solution
Example Calculation
Distance from Point to xy-plane
Problem: Find distance from point (1, 1, 1) to the plane x + y = 0
Point: (1, 1, 1)
Plane equation: x + y + 0z + 0 = 0 (so A=1, B=1, C=0, D=0)
Solution
Step 1: Apply formula: d = |Aa + Bb + Cc + D| / √(A² + B² + C²)
Step 2: Substitute: d = |1(1) + 1(1) + 0(1) + 0| / √(1² + 1² + 0²)
Step 3: Calculate: d = |1 + 1 + 0 + 0| / √(1 + 1 + 0) = |2| / √2
Step 4: Simplify: d = 2 / √2 = √2 ≈ 1.414
Result: Distance = √2 ≈ 1.414 units
Distance Formulas
Standard Form
d = |Aa+Bb+Cc+D|/√(A²+B²+C²)
For plane Ax + By + Cz + D = 0
Normal Vector Form
d = |A(a-x)+B(b-y)+C(c-z)|/√(A²+B²+C²)
With normal [A,B,C] and point (x,y,z)
Special Cases
xy-plane: d = |c|
xz-plane: d = |b|
yz-plane: d = |a|
Quick Tips
Distance is always the shortest (perpendicular) path
A, B, and C cannot all be zero for a valid plane
The formula works for any orientation in 3D space
Distance is always positive (absolute value)
Use normal vector form when you know a point on the plane
Understanding Distance from Point to Plane
What is Point-to-Plane Distance?
The distance from a point to a plane is the length of the perpendicular line segment from the point to the plane. This is always the shortest possible distance between the point and any point on the plane.
Real-World Applications
- •Architecture and construction (clearance calculations)
- •Computer graphics and 3D modeling
- •Aerospace engineering (trajectory calculations)
- •Physics simulations and collision detection
Formula Derivation
Standard Form:
d = |Aa + Bb + Cc + D| / √(A² + B² + C²)
This formula comes from the concept that the distance is the magnitude of the projection of the vector from any point on the plane to the given point onto the normal vector of the plane.
Key Properties
- Perpendicular: Always the shortest distance
- Unique: Only one perpendicular from point to plane
- Sign-independent: Distance is always positive
- Scale-invariant: Multiplying plane equation by constant doesn't change distance
Remember: If A² + B² + C² = 0, then the equation doesn't represent a plane in 3D space, making the distance calculation undefined.
Types of Plane Equations
Standard Form
Ax + By + Cz + D = 0
- • Most common representation
- • Direct use in distance formula
- • [A, B, C] is the normal vector
- • Easy to identify plane orientation
Normal Vector & Point
Normal [A,B,C] through point (x₀,y₀,z₀)
- • Intuitive geometric representation
- • Useful when you know a point on plane
- • Can convert to standard form
- • Common in 3D modeling software
Special Cases and Examples
Coordinate Planes
- xy-plane: z = 0, distance = |c|
- xz-plane: y = 0, distance = |b|
- yz-plane: x = 0, distance = |a|
Origin Distance
Distance from plane to origin (0,0,0):
d = |D| / √(A² + B² + C²)
Parallel Planes
For planes Ax+By+Cz+D₁=0 and Ax+By+Cz+D₂=0:
d = |D₂-D₁| / √(A² + B² + C²)