SOLID ANGLE, an angle formed by three or more plane angles in a point.
The sum of all the plane angles constituting a solid angle, is always less than 300°; otherwise they would constitute the plane of a circle, and not a solid.
If the apex of a solid angle be supposed in the centre of a sphere, the measure of the solid angle is the space inter cepted upon the surface of the sphere by the planes of such angle.
Hence the comparison of solid angles is easily effected : for since the areas of spherical triangles are measured by the excess of the sums of their angles, each above two right angles, and the areas of spherical polygons of n sides by the excess of the sum of their angles above (2 n-4) right angles ; it follows, that the magnitude of a trilateral solid angle will be measured by the excess of the sum of three angles above (2.3-4=2) above two right angles ; and the magnitude of solid angles, formed by n bounding planes, will be measured by the excess of the sum of the angles of inclination of the several planes above (2 n-4) right angles.
In all cases, the maximum limit of solid angles will be the plane, towards which various planes determining such angles approach, as they diverge farther from each other about the same summit; the same as a right line is the maximum limit of plane angles, being formed by the two boundary lines, when they make an angle of 180°. The maximum limit of solid angles is measured by the surface of the hemisphere, in like manner as the maximum limit of plane angles is measured by the are of a semicircle.
The solid right angle is 1=0-)h of the maximum solid angle ; while the plane right angle is half the maximum plane angle.