Solid and Dihedral Angles of a Tetrahedron

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During an eclipse, the Moon and Sun appear to have roughly the same size viewed from Earth. The field of view of an object is called the solid angle. A unit sphere around the observer has a solid angle of steradians, the same as the surface area.


Let be a tetrahedron and consider the solid angles defined by an observer at each vertex looking at its opposite triangle. You might think that the smallest solid angle corresponds to the smallest of the four triangles, but that is not necessarily so—it depends on the inclination of the triangle relative to the observer at the vertex.

When two planes intersect, the angle between them is called the dihedral angle. In a cube, the dihedral angles are (or 90°). The maximum possible dihedral angle is .

For a triangle in the plane, let be an interior angle. Then , the full circle.

For a tetrahedron in 3D space, let be a dihedral angle and be a solid angle. Then , the full sphere.


Contributed by: Ed Pegg Jr (July 2018)
Open content licensed under CC BY-NC-SA



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