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Why does the formula $$\Omega =( \text{angular diameter} * \frac{\pi}{180^{\circ}})^2 * \frac{\pi}{4}$$ give a more accurate calculation of the solid angle of a light source from its angular diameter than the formula $$\Omega =( \text{angular diameter} * \frac{\pi}{180^{\circ}})^2$$

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It is just the formula $A=\pi r^2=\pi d^2/4$ for the area of a circle in terms of its diameter. Think of the angular diameter as a length, and the solid angle as an area (both suitably normalized by the distance of the object from the point of observation). The light source is supposed to be circular in appearance, unless there is a good reason to suggest otherwise.

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