A generalized framework from HCR's Theory of Polygon is presented for computing the solid angle subtended by an arbitrary polygonal plane, regular or irregular, at any point in three-dimensional space. The approach is unified and systematic, relying on a single master formula derived for a right triangular plane. This formula is simplified and equivalently expressed in terms of inverse trigonometric functions, including arcsine, arccosine, and arctangent. The variation of the solid angle with respect to the orthogonal sides of the triangle and the distance of the observation point is illustrated graphically. In addition, several corollaries are established for the solid angle subtended by planar surfaces, both polygonal and non-polygonal, at different coplanar locations of the observation point. The results are derived using the standard formula for right-triangle geometry and the concept of the angle of vision for observation of two-dimensional figures.