Draw a circle around a polygon, and call its radius "1 radian". Drop perpendiculars from the center of the circle through each side. Pairs of adjacent perpendi

Draw a circle around a polygon, and call its radius "1 radian". Drop perpendiculars from the center of the circle through each side. Pairs of adjacent perpendiculars each subtend an arc of the circle, of length equal to the "exterior angle" between the corresponding sides. Those arcs sum to 2π radians. The intersection points with the circle from a sort of "dual" polygon to the original, exchanging sides with vertices, and exchanging central angles with exterior angles. (But this dual construction is not uniquely reversible back to a non cyclic polygon or not using the same center) Now take a polyhedron and choose a center and draw a sphere around it. Drop perpendiculars to faces, intersecting with the sphere to create new "dual" vertices from faces. This time, draw edges between adjacent-face perpendiculars , so each original edge generates a new dual (perpendicular!) edge. This now creates a new face for each vertex of the original shape. Now we have a dual polyhedron inscribed in a sphere, where each face subtends the 2D "exterior angle" of an vertex of the original polygon. Note that an exterior (hyper!)angle of a hyperpolyhedron in N-dimensional space is an N-1 dimensional measure (face-dimensional dimensional measure) of a vertex (with its cluster of incident faces). In 2D, there is an isomorphism between pairs-of-edges and ordinary 1D angles. (and edges are 2D hyper-faces). But in higher dimensions we need use N-1 dimensional dual hyperfaces to measure N-1 dimensional exterior angles I'm not actually sure this really works in n>3 dimensions. Thinking about HN favorite Geometric Algebra is a little helpful for the intuition here. For concave shapes, some of these angles are negative, as the relevant side/face is oriented the opposite way from its neighbors, with respect to the enclosing circle/sphere/hypersphere. These N-1 dimensional angles in N dimensional are called "solid" angles. The unit is the "steradian", which is usually considered dimensionless, but cool cats know that its dimension is really the dimension of normalized hypersurface hyperarea. (Similar to how a https://en.m.wikipedia.org/wiki/Solid_angle https://en.m.wikipedia.org/wiki/Solid_angle