Angle Between Two Planes Calculator

Work out angle between two planes instantly with clear inputs, formula shown and shareable results.

About Angle Between Two Planes Calculator

The angle between two planes equals the angle between their normal vectors: cos θ = |n₁·n₂| / (|n₁||n₂|). The absolute value keeps the answer in the range 0° to 90°, the usual convention for a dihedral angle.

Formula

Angle between planes

cos θ = |n₁ · n₂| / (|n₁| |n₂|)

Worked example

With the default values below, Angle Between Two Planes Calculator returns Dihedral angle of 63.6122. Change any input above and every figure here is recalculated in your browser.

Inputs used

Normal of plane 1 (a, b, c): 1, 2, 2
Normal of plane 2 (a, b, c): 2, -1, 2

Results

Dihedral angle: 63.6122 °
cos of the angle: 0.44444444
Angle in radians: 1.110242
Relationship: Oblique planes

Step by step

1Values used
2Angle between planes
3Dihedral angle= 63.6122 °
4cos of the angle= 0.44444444
5Angle in radians= 1.110242
6Relationship= Oblique planes

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For normals (1,2,2) and (2,−1,2)?

The dot product is 4, both magnitudes are 3, so cos θ = 4/9 and θ ≈ 63.6122°.

Why use the absolute value?

Flipping a normal's direction does not change the plane, so without the absolute value you could get the supplement of the intended angle.