Point to Plane Distance Calculator

Distance from a Point to a Plane in 3D

This calculator finds the shortest (perpendicular) distance from a point in three-dimensional space to a plane given in the standard form Ax + By + Cz + D = 0. It is useful for 3D geometry homework, physics problems, engineering design checks, and computer graphics or robotics tasks.

On this page you will find the point-to-plane distance formula, a step-by-step numerical example, guidance on how to interpret the result, and a short list of assumptions and limitations for the calculator.

Point to Plane Distance Formula

Consider a point in 3D space and a plane with equation Ax + By + Cz + D = 0, where A, B, and C are the components of the plane’s normal vector.

The perpendicular distance from the point to the plane is given by:

In a more compact text-only form, the same formula is:

d = |A·x₀ + B·y₀ + C·z₀ + D| / √(A² + B² + C²)

Here:

• (x₀, y₀, z₀) are the coordinates of the point.
• A, B, C, D are the coefficients in the plane equation.
• The absolute value in the numerator ensures the distance is always non-negative.
• The square root in the denominator is the magnitude of the normal vector (A, B, C).

How the Formula Works (Conceptual Overview)

The plane Ax + By + Cz + D = 0 has a normal vector n = (A, B, C), which is perpendicular to every direction lying inside the plane. To measure how far a point is from the plane, you only care about separation along this normal direction.

If you substitute the point into the plane equation, you obtain the value:

S = A·x₀ + B·y₀ + C·z₀ + D

This quantity tells you how far the point is from satisfying the plane equation. Dividing S by the length of the normal vector ‖n‖ = √(A² + B² + C²) converts that raw value into an actual geometric distance. Taking the absolute value removes the sign and leaves only the magnitude (the shortest distance), which is what this calculator returns.

How to Interpret the Result

After entering the point coordinates and plane coefficients into the calculator, the result is a single non-negative number representing the perpendicular distance from the point to the plane.

• Distance = 0: the point lies exactly on the plane, within numerical precision.
• Small distance: the point is very close to the plane, which can indicate a near contact or small clearance in engineering or graphics applications.
• Larger distance: the point is farther away from the plane along the direction normal to the plane.

The calculator does not show whether the point is on a particular side of the plane. That information is contained in the sign of A·x₀ + B·y₀ + C·z₀ + D, but the distance uses the absolute value and therefore discards the sign.

Units are preserved: if you enter coordinates and plane parameters in meters, the resulting distance is in meters; if you use centimeters, the result is in centimeters, and so on.

Worked Example: Distance from a Point to a Plane

Suppose you want the distance from the point (1, 2, 3) to the plane 2x − y + 2z − 4 = 0.

1. Identify the coefficients and point coordinates:

   • A = 2
   • B = −1
   • C = 2
   • D = −4
   • x₀ = 1, y₀ = 2, z₀ = 3

2. Compute the numerator |A·x₀ + B·y₀ + C·z₀ + D|:

   A·x₀ + B·y₀ + C·z₀ + D = 2·1 + (−1)·2 + 2·3 + (−4)

   Step by step:

   • 2·1 = 2
   • (−1)·2 = −2
   • 2·3 = 6
   • Sum: 2 − 2 + 6 − 4 = 2

   Therefore the numerator is |2| = 2.

3. Compute the denominator √(A² + B² + C²):

   A² + B² + C² = 2² + (−1)² + 2² = 4 + 1 + 4 = 9

   So the denominator is √9 = 3.

4. Divide numerator by denominator:

   d = 2 / 3 ≈ 0.6667

The perpendicular distance from the point (1, 2, 3) to the plane 2x − y + 2z − 4 = 0 is therefore 2/3 units, or about 0.67 in decimal form.

You can reproduce this example in the calculator by entering:

• Point x = 1
• Point y = 2
• Point z = 3
• Plane A = 2
• Plane B = −1
• Plane C = 2
• Plane D = −4

Special Cases and Comparison

The same distance formula behaves in a straightforward way in several important special cases. The table below compares some typical setups and what the distance formula reduces to.

Assumptions, Limitations, and Edge Cases

This calculator focuses on the perpendicular distance from a point to a plane. To interpret its output correctly, keep the following assumptions and limitations in mind.

Assumptions

• Valid plane definition: the coefficients A, B, and C are not all zero at the same time. If A = B = C = 0, the expression Ax + By + Cz + D = 0 does not define a plane, and the distance formula breaks down.
• Real-valued inputs: all coordinates and plane coefficients are treated as real numbers. Complex-valued geometry is not supported.
• Consistent units: the plane equation and point coordinates must use the same length units. Mixing units (for example, meters in the plane equation and centimeters for the point) will make the numerical distance meaningless.

Limitations

• Unsigned distance only: the tool returns the magnitude of the distance. It does not report a signed distance, which would indicate on which side of the plane the point lies.
• No nearest-point coordinates: the calculator does not currently compute the coordinates of the closest point on the plane. It only outputs the scalar distance.
• Numerical precision: for very large or very small input values, floating-point rounding errors may cause extremely small non-zero distances for points that are theoretically on the plane.
• No symbolic manipulation: the inputs are numeric only; the tool does not simplify symbolic plane equations or algebraic expressions.

The underlying distance formula is standard in analytic geometry and multivariable calculus texts, where it is usually derived using vector projections onto the plane’s normal vector. Many textbooks on linear algebra and vector calculus present essentially the same expression for the distance from a point to a plane.