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The lead states "For a given shape, SA:V decreases linearly with increasing size". This is only true given some assumptions on the geometry. Fractals are an example where this statement is not as trivial, see fractal dimension. --Berland (talk) 19:05, 3 June 2009 (UTC)
The associated page contains a mistake, or at least a misleading presentation. For instance, in the table regarding SA:V, for the cube, the variable "a" denotes a diameter. Yet, in the case of the sphere, it denotes a radius. The SA:V of cube and sphere are identical, 3/r, where r is the radius or, in the case of the cube, the half-side, or the length of the perpendicular projection of a normal from a side going through the cube's center.
The comparison of cube to sphere or sphere to any other shape should use an equivalent radius. This radius can be derived either by equating the areas of the two solids, obtaining the radius, and then calculating the surface-area-to-volume ratio using it, or by equating the volumes of the two solids, obtaining the radius, and then calculating. If, for instance, the area of a sphere is equated to the area of a cube, the equivalent radius is sqrt(3/(2*pi))*a, using the nomenclature on the page. The corresponding SA:V ratio is sqrt(6*pi)/a. If the volume of a sphere is equated to the volume of a cube, the equivalent radius is a times cube root of 3/(4*pi), and the corresponding SA:V is a divided into the cube root of 4/(3*pi).
The graph is misleading, using length as the x axis is not the best choice, since a dodecahedron with a side length of 1 is much larger in volume than a sphere with a radius of 1, and the SA/V ratio is size-dependent. Perhaps the graph should be remade using volume on the x axis to give a better intuitive representation. If, as I suspect, this doesn't generate a lot of discussion, I may just do that. Drunaii (talk) 04:48, 17 February 2012 (UTC)
^ Just to say, I would second the above suggestion; the graph as-is can easily be misread to incorrectly imply that a dodecahedron is a more efficient SVR packing shape than a sphere! Should be re-done — Preceding unsigned comment added by 18.104.22.168 (talk) 11:16, 2 May 2013 (UTC)
- Support — the graph is misleading because "length" has a different meaning in the sphere. A much better graph is surface area vs volume. I'll draw that tonight... cmɢʟee୯ ͡° ̮د ͡° ੭ 19:15, 12 June 2013 (UTC)
Found how to derive a cone's surface equation at http://www.mathopenref.com, the equation for the slant height of a cone s=(r^2 + h^2)^1/2, allows for the ratio of circumferences 2*pi*r over 2*pi*s, to equate to, the unknown area of the cone's top surface over the known area of the larger circle pi*s^2, giving the cone's top surface pi*r*s, the cone's total area = pi*r*s + pi*r^2. To find its volume see, Volume of a Pyramid and a cone : http://nrich.maths.org , a cube made from three Yangma pyramids, is used to prove that a cone's volume =(pi*r^2)h/3. Ratio, SA:V =[3*(s+r)/rh]. For proof of the equations of the truncated cone, see, http://www.vitutor.com/geometry/solid/truncated_cone.html, for cones with domes, see, Spherical Cone at wolfram.com. While using Google Search, to plot the graph of the cone's, SA:V ratio where height and radius are equal, to my surprise it plotted their negative values as well, cone shaped holes also have surfaces and volumes. Cuberoottheo (talk) 18:09, 27 September 2014 (UTC)