Finding the Surface Area and Volume of a Cuboid in One Step

摘要

Now if e = 0, this just gives 140 which is the volume of our cuboid. This corresponds to the yellow part. To clad our cuboid with, say, some insulation of thickness ?, we need the surface area times e, this gives the cyan parts whose total volume is 166e. Then along the edges we will need strips of insulation e by e with a total length which is the sum of the edges of the box, shown red in the diagram. The 166 therefore represents the total surface area, and the 64 represents the total edge length of the cuboid. The last term in e3 comes from needing small cubes of side e at each of the corners. We therefore see the 8 represents the number of corners or vertices, shown brown in the diagram.