Dihedral angle and its linear angle
Take a line m in space. Two half-planes that start on m make a dihedral angle. The line m is its edge and the half-planes are its faces. Think of an open book: the spine is the edge, the covers are the faces.
How do we give it a number? Pick any point P on the edge. In each face draw the ray from P that is perpendicular to the edge. The angle between these two rays is the linear angle. The size of the dihedral angle is the size of its linear angle.
Why does this work? Move P along the edge. The two rays slide along but keep the same angle between them. So the number does not depend on P.
- A dihedral angle is between 0° and 180°.
- If it is 90°, the two planes are perpendicular.
- Two planes that cross make four dihedral angles. Opposite ones are equal.
Trihedral angle: face angles and dihedral angles
Take three rays OA, OB, OC from one point O, not all in one plane. The three flat regions between pairs of rays make a trihedral angle. O is the vertex, the rays are the edges, the three flat regions are the faces.
A trihedral angle carries six angles:
- Three face angles (plane angles) α, β, γ. They are ordinary flat angles, one on each face. α is on the face opposite edge 1, β opposite edge 2, γ opposite edge 3.
- Three dihedral angles A, B, C, one at each edge. A is at edge 1, B at edge 2, C at edge 3. So A is opposite α, B opposite β, C opposite γ.
Which corners can exist? Each face angle must be smaller than the sum of the other two, and α + β + γ must be less than 360°. Try to break either rule in the 3D slider: a red warning appears.
A cube corner has α = β = γ = 90° and A = B = C = 90°. A regular tetrahedron corner has α = β = γ = 60°, but A = B = C ≈ 70.5°.
Polyhedral angle and the 360° rule
Use more than three rays and you get a polyhedral angle (for example four faces at the tip of a square pyramid). A polyhedral angle is convex if it stays on one side of each of its face planes.
Rule: the face angles of a convex polyhedral angle add to less than 360°.
Why? Lay the corner flat by pressing it down. The faces would just fit round the vertex when the sum is 360°, and then there is no corner left, only a flat sheet. Any real corner has a gap to close, so its sum is smaller.
- Cube corner: 3 × 90° = 270°.
- Regular tetrahedron corner: 3 × 60° = 180°.
- Regular dodecahedron corner: 3 × 108° = 324°.
- Four equilateral triangles round a point (square pyramid): 240°.
This is also why only five regular polyhedra exist: you need at least three faces at a vertex and the angle sum must stay below 360°.
Cosine and sine theorems for a trihedral angle
Cosine theorem for face angles: cos α = cos β · cos γ + sin β · sin γ · cos A.
Cosine theorem for dihedral angles: cos A = −cos B · cos C + sin B · sin C · cos α.
Sine theorem: sin α / sin A = sin β / sin B = sin γ / sin C.
The same shape repeats by swapping letters: (α, A), (β, B), (γ, C) go round together.
Where does the first rule come from? Take unit vectors e₁, e₂, e₃ along the three edges. Angle between e₂ and e₃ is α, so e₂ · e₃ = cos α. Split e₂ = (cos γ) e₁ + p and e₃ = (cos β) e₁ + q, where p and q are perpendicular to e₁. Their lengths are sin γ and sin β, and the angle between p and q is the dihedral angle A at edge e₁. Then cos α = cos γ cos β + p · q = cos β cos γ + sin β sin γ cos A.
Handy form: if you know the three face angles, cos A = (cos α − cos β cos γ) / (sin β sin γ). If you know the three dihedral angles, cos α = (cos A + cos B cos C) / (sin B sin C). The dihedral angles of a real corner always satisfy A + B + C > 180°.
Try it: build and measure a corner
With paper. Fold a sheet in half. Stand it on a table and open it. On each half draw a line from the fold at 90° to the fold. Use a protractor on the two lines: that is the dihedral angle. Open it more and measure again.
In the 3D. Predict first: if you set α = β = γ = 90°, what will A, B and C be? Then check. Next make α = β = 90° and slowly change γ. Which dihedral angle follows γ?
Key formulas and definitions
- Dihedral angle = its linear angle (rays ⊥ edge, one in each face)
- Trihedral angle: face angles α, β, γ; dihedral angles A, B, C (A opposite α, ...)
- cos α = cos β cos γ + sin β sin γ cos A
- cos A = − cos B cos C + sin B sin C cos α
- sin α / sin A = sin β / sin B = sin γ / sin C
- Existence: each face angle < sum of the other two; α + β + γ < 360°; A + B + C > 180°
- Convex polyhedral angle: sum of face angles < 360°
Worked examples
1. A box corner has three face angles of 90°. Find the dihedral angle at one edge.
cos A = (cos 90° − cos 90° cos 90°) / (sin 90° sin 90°) = (0 − 0) / 1 = 0. So A = 90°. All three dihedral angles are right angles, which is why the faces of a box are perpendicular.
2. Three equilateral triangles meet at a vertex of a regular tetrahedron. Find the angle between two faces.
α = β = γ = 60°. cos A = (cos 60° − cos² 60°) / sin² 60° = (0.5 − 0.25) / 0.75 = 1/3. So A = arccos(1/3) ≈ 70.53°.
3. Can three flat angles of 100°, 130° and 140° form a trihedral angle?
Their sum is 370°, which is more than 360°. So no corner can be built.
4. Can face angles of 40°, 50° and 100° form a trihedral angle?
The sum is 190°, below 360°, but 100° is bigger than 40° + 50° = 90°. One face angle is too large, so the corner cannot exist.
5. Three regular pentagons (each angle 108°) meet at a vertex of a dodecahedron. Find the angle between two pentagon faces.
cos 108° ≈ −0.3090, cos² 108° ≈ 0.0955, sin² 108° ≈ 0.9045. cos A = (−0.3090 − 0.0955) / 0.9045 ≈ −0.4472. So A ≈ 116.57°.
6. The three dihedral angles of a trihedral corner are each 80°. Find each face angle.
cos α = (cos A + cos B cos C) / (sin B sin C) = (0.1736 + 0.0302) / 0.9698 ≈ 0.2101. So α ≈ 77.9°. (Check: 80° × 3 = 240° > 180°, as needed.)
Common mistakes
- Measuring the dihedral angle with any two lines. The two rays must both be perpendicular to the edge.
- Mixing up the face angle (flat, on one face) with the dihedral angle (between two faces).
- Using the wrong pair in the formula. In cos α = cos β cos γ + sin β sin γ cos A, the angle A is the one opposite α.
- Allowing α + β + γ = 360° or more. At 360° the corner is completely flat, so it is not a corner.