02-15-2014, 05:29 AM
A few days earlier I made a formula to determine the compactness of a rectangular prism. Now I felt that compactness had more to do than just volume because a 10x10x10 cube would be more compact than a 1x1x1000 rectangular prism, even though they have the same volume. So I based it off of how close to a cube the shape was, since a cube is the most compact rectangular prism you can have. It is: (3*cbrt(l*w*h))/(l+w+h). It's fairly simple to use. Here is how I created this formula:
I wanted to base it off of the side lengths. The l*w*h factor is the volume of the rectangular prism. Taking the cube root of that gives what the side lengths would be if it were a cube. the (l+w+h)/3 factor is the average of the side lengths. if you take the cbrt(l*w*h) and divide that by (l+w+h)/3, you get the same as cbrt(l*w*h)*((l+w+h)/3), or just (3*cbrt(l*w*h))/(l+w+h).
If the result is 1, the rectangular prism is a perfect cube. If it is less than one, than it is not a cube. The closer to 1 the number is, the more compact it is.
I wanted to base it off of the side lengths. The l*w*h factor is the volume of the rectangular prism. Taking the cube root of that gives what the side lengths would be if it were a cube. the (l+w+h)/3 factor is the average of the side lengths. if you take the cbrt(l*w*h) and divide that by (l+w+h)/3, you get the same as cbrt(l*w*h)*((l+w+h)/3), or just (3*cbrt(l*w*h))/(l+w+h).
If the result is 1, the rectangular prism is a perfect cube. If it is less than one, than it is not a cube. The closer to 1 the number is, the more compact it is.