2 days ago
- The volume-to-surface-area ratio for a sphere is V/A = r/3, where r is the radius.
- The same ratio r/3 holds for all regular solids when r is the radius of the inscribed sphere.
- For a cube with edge a, r = a/2, giving volume 8r³, area 24r², and ratio r/3.
- Other regular solids (tetrahedron, octahedron, dodecahedron, icosahedron) have more complex relationships but still yield the ratio r/3.
- Proof: By forming pyramids from each face to the center, volume = fB r/3 and area = fB, leading to ratio r/3.
- The theorem generalizes to n dimensions: the ratio of n-dimensional volume to (n-1)-dimensional boundary volume is r/n.
- The derivative of volume with respect to r gives the surface area, a classic relationship.