Prism properties, V=volume, A=total surface area, x, y, z are length, width and depth (any order).
V=xyz, A=2(xy+xz+yz)
To find x, y, or z, or any pair of these we can use the table (Y means yes, we have the value, N means no, we don’t):
V A x y MEANING
1 N Y Y Y Given A and two linear dimensions, can find z
2 Y N Y Y Given V and two linear dimensions, can find z
3 Y Y Y N Given A and V and one linear dimension, can find y and z
Solution for each condition:
1 (Given A and two linear dimensions, can find z)
A=2(xy+xz+yz)=2xy+2xz+2yz, A-2xy=z(2x+2y), z=(A-2xy)/(2x+2y).
Example: if we have the total surface area and length and breadth we can find the depth. In this case, z is the depth and x and y are length and breadth.
2 (Given V and two linear dimensions, can find z)
V=xyz, z=V/(xy).
Example: if we have the volume of the prism (cuboid) and breadth and depth, we can find the length. In this case, z is the length and x and y are breadth and depth.
3 (Given A and V and one linear dimension, can find y and z)
This is more complicated. All we have in the linear dimension is x (length, breadth or depth) and we have to find the other two linear dimensions using A and V.
yz=V/x, A=2(xy+xz+yz)=2(xy+xz+V/x). But xz=V/y, so:
A=2(xy+V/y+V/x). We know A, V and x so we can find y.
Multiply through by xy:
Axy=2(x²y²+Vx+Vy). 2x²y²+2Vx+2Vy-Axy=0 which is a quadratic in y.
Rewrite: 2x²y²+(2V-Ax)y+2Vx=0.
Quadratic formula:
y=(Ax-2V±√((Ax-2V)²-16Vx³))/(4x²), giving us two values for y. The two solutions should correspond to the values for y and z.
To find z we use z=V/(xy) using each value for y we found. We should find that the values of y and z correspond to the quadratic solutions.
Example: We are given the depth, V and A, so we can find the length and breadth. In this case, the depth is x, and y and z are the length and breadth.
The depth can be any of x, y or z so you can use whatever formula above fits your particular problem. Make a note of what you are given from the table. Then find out which line of the table fits your problem. Follow the solution for that line substituting the values you have been given.