Calculator Inputs
Enter each plane in the form Ax + By + Cz + D = 0.
Example data table
| Plane 1 | Plane 2 | Normal vectors | Acute angle | Relation |
|---|---|---|---|---|
| x + 2y + 2z - 5 = 0 | 2x - y + 2z + 3 = 0 | (1,2,2) and (2,-1,2) | 63.6122° | Intersecting planes |
| x + y + z - 6 = 0 | x + y + z - 2 = 0 | (1,1,1) and (1,1,1) | 0° | Parallel planes |
| x + y - 3 = 0 | x - y + 1 = 0 | (1,1,0) and (1,-1,0) | 90° | Perpendicular intersecting planes |
| 2x + 4y + 6z - 8 = 0 | x + 2y + 3z - 4 = 0 | (2,4,6) and (1,2,3) | 0° | Coincident planes |
These examples show acute, perpendicular, parallel, and coincident cases.
Formula used
For planes
their normal vectors are
The acute angle between the planes is
where
The cross product n₁ × n₂ gives the intersection-line direction when the planes are not parallel.
How to use this calculator
1. Enter A, B, C, and D for the first plane.
2. Enter A, B, C, and D for the second plane.
3. Choose decimal precision and your preferred output unit.
4. Click Calculate Angle to generate the result.
5. Review the acute angle, supplementary angle, and plane relation.
6. Inspect the dot product, magnitudes, and intersection direction.
7. Use the Plotly graph to compare both normal vectors visually.
8. Export the result summary using CSV or PDF buttons.
Frequently asked questions
1. Why does the calculator use normal vectors?
A plane’s orientation is determined by its normal vector. The angle between two planes is therefore obtained from the angle between their normals, using the acute version for standard geometry problems.
2. Why is the absolute value used in the formula?
The absolute value ensures the reported plane angle is acute. Two planes share the same acute angle even when the normals point in opposite directions.
3. What happens when both planes are parallel?
Parallel planes have proportional normal vectors, so the acute angle becomes 0°. They may still be distinct planes or the exact same plane, depending on all coefficients.
4. What does a 90° result mean?
A 90° acute angle means the planes are perpendicular. Their normal vectors have a dot product of zero, and the planes intersect at right angles.
5. Does the D term affect the angle?
No. The angle depends only on the normal vectors, so only A, B, and C affect it. D shifts the plane without changing orientation.
6. What does the cross product tell me?
The cross product of the two normals gives a direction vector for the line where the planes intersect. If that vector becomes zero, the planes are parallel.
7. Can I use decimal coefficients?
Yes. The calculator accepts integers and decimals. It also lets you choose the displayed precision for cleaner reporting and exports.
8. Why are both radians and degrees shown?
Many geometry courses use degrees, while advanced mathematics often uses radians. Showing both avoids repeated conversions and supports mixed workflows.