Dimitris M. Christodoulou,
Euclidean figures and solids without incircles or inspheres,
Forum Geometricorum, 16 (2016) 291--298.
Abstract. All classical convex planar Euclidean figures that possess incircles have areas A=pr/2, where p is the perimeter, r is the radius of the incircle, and the factor of 2 represents the dimension of the space. Similarly, all classical convex Euclidean solids that possess inspheres have volumes V=Sr/3, where S is the total surface area, r is the radius of the insphere, and the factor of 3 represents the dimension of the space. Elementary figures such as parallelograms and trapezoids without an incircle still obey the same area relation, but then r is the harmonic mean of the radii of the two internally tangent circles to opposite sides. Similarly, common solids without an insphere (notably cylinders and prisms) still obey the same volume relation, but then r is the harmonic mean of the three internally tangent spheres to their faces.
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