1

We know that the sum of angles of a triangle equals the straight angle (180 degrees).

Can we convert a 2D theorem to 3D?

e. g. We can generalize the triangle to a tetrahedron, angles of the triangle to the dihedral angles of tetrahedron. Do we know or can we calculate the sum of dihedral angles of the tetrahedron?

gt6989b
  • 54,422
Paull
  • 55
  • it's not very clear how you would define angles in 3D? Usual 2D angles are formed by intersection of 2 lines at a point. You need to intersect 3 planes to get a vertex of the tetrahedron, so what would be the measure of the resulting "angle"? – gt6989b May 23 '22 at 18:56
  • 1
    A regular tetrahedron has six dihedral angles of $\cos^{-1}(\frac13)$ which is not a nice number. I doubt this stays constant for irregular tetrahedra. – Henry May 23 '22 at 18:59
  • @gt6989b In 2D we consider interior angles of the triangle, so in 3D we can generalize "interior" angles of tetrahedron to dihedral angles. – Paull May 23 '22 at 19:07
  • 2
    @gt6989b There exist two generalization of angle to 3D. Dihedral angles and solid angles. – jjagmath May 23 '22 at 19:08
  • @Henry maybe we can estimate it? – Paull May 23 '22 at 19:08
  • @Paull Apparently the sum of the dihedral angles is between $2\pi$ and $3\pi$ (or $360^\circ$ and $540^\circ$ in degrees) – Henry May 23 '22 at 19:13

1 Answers1

3

The similar theorem for tetrahedrons is that the sum of its solid angles plus twice the sum of its dihedral angles is $4\pi$

See here

jjagmath
  • 18,214
  • Thanks! Now I know that I can generalize 2D theorem to 3D in this way: Sum of solid angles of tetrahedron is a half of the maximum solid angle that can be subtended at any point (4π sr.); like in 2D: half of round angle. Later i can check if this statements its true. – Paull May 23 '22 at 19:58
  • I found out that the sum of the four solid angles of a tetrahedron is less than 2π. Is there a simple proof of this? – Paull May 23 '22 at 19:59