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If the legs have lengths a, b, c, then the trirectangular tetrahedron has the volume [2] =. The altitude h satisfies [3] = + +. The area of the base is given by [4] =. The solid angle at the right-angled vertex, from which the opposite face (the base) subtends an octant, has measure π /2 steradians, one eighth of the surface area of a unit sphere.
Picture Name Schläfli symbol Vertex/Face configuration exact dihedral angle (radians) dihedral angle – exact in bold, else approximate (degrees) Platonic solids (regular convex)
The 6 edge lengths - associated to the six edges of the tetrahedron. The 12 face angles - there are three of them for each of the four faces of the tetrahedron. The 6 dihedral angles - associated to the six edges of the tetrahedron, since any two faces of the tetrahedron are connected by an edge.
What condition on 12 angles is necessary and sufficient for them to be the 12 angles of some tetrahedron? Clearly the sum of the angles of any side of the tetrahedron must be 180°. Since there are four such triangles, there are four such constraints on sums of angles, and the number of degrees of freedom is thereby reduced from 12 to 8. The ...
Fig. 3 – Applications of the law of cosines: unknown side and unknown angle. Given triangle sides b and c and angle γ there are sometimes two solutions for a. The theorem is used in solution of triangles, i.e., to find (see Figure 3): the third side of a triangle if two sides and the angle between them is known: = + ;
Two sides and the included angle (SAS, side-angle-side) Two sides and an angle not included between them (SSA), if the side length adjacent to the angle is shorter than the other side length. A side and the two angles adjacent to it (ASA) A side, the angle opposite to it and an angle adjacent to it (AAS).
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When longitude spans 2 π radians and latitude spans π radians, the solid angle is that of a sphere. A latitude-longitude rectangle should not be confused with the solid angle of a rectangular pyramid. All four sides of a rectangular pyramid intersect the sphere's surface in great circle arcs. With a latitude-longitude rectangle, only lines of ...