The sphere of the radis R touches the faces of the dihedral angle, the value of which is alpha

The sphere of the radis R touches the faces of the dihedral angle, the value of which is alpha. Determine the distance from the center of the sphere to the edge of the dihedral.

Let us construct from point O the radii OB and OC to the points of tangency.

The radii drawn to the points of tangency are perpendicular to the tangents themselves, then the triangles AOB and AOC are rectangular.

In triangles AOB and AOC, the hypotenuse of AO is common, leg OB = OC = R, then the triangles are equal in leg and hypotenuse, which means the angle OAC = OAB = BAC / 2 = α / 2.

In a right-angled triangle OAB Sin (α / 2) = R / OA.

OA = R / Sin (α / 2) see

Answer: The distance from the center of the sphere to the edge R / Sin (α / 2) cm.