Platonic solid the inradius "r". That is, The following table lists the various radii of the Platonic solids together with their surface area and volume. The overall size is fixed by taking the edge length, "a", to be equal to 2. The constants "φ" and "ξ" in the above are given by Among the Platonic solids, either the dodecahedron or the icosahedron may be seen as the best approximation to the sphere. The icosahedron has the largest number of faces and the largest dihedral angle, it hugs its inscribed sphere the most tightly, and its surface area to volume ratio is closest
Gabriel's Horn using Cavalieri's principle before the invention of calculus, but today calculus can be used to calculate the volume and surface area of the horn between and , where . Using integration (see Solid of revolution and Surface of revolution for details), it is possible to find the volume and the surface area : The surface area formula above gives a lower bound for the area as 2 times the natural logarithm of . There is no upper bound for the natural logarithm of as approaches infinity. That means, in this case, that the horn has an infinite surface area. That