Suppose I have a rectangular pyramid. I partition the dihedral angle between a fixed pair of opposite faces into three parts and thereby obtain three sub-pyramids (within the original one). Consider the sum of the solid angles subtended at the vertex by these sub-pyramids. Is it equal or not to the solid angle subtended by the original pyramid at the vertex ? Please explain.
2026-03-13 16:31:51.1773419511
Solid angle of a pyramid
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The sum of the solid angles is equal to the total solid angle. The apparent contradiction with your formula follows from the fact that it only works for a straight pyramid, and your subpyramides are not straight. Observe that an oblique pyramid with the same height and base subtends a smaller solid angle. The more oblique, the lesser the angle.