Two Parcels

Two competitive children get delivered presents. Both presents take the form of wrapped cuboid boxes. Jessica gets a tape measure and wraps it around her present. She smugly declares that it is 60” around. Zhenya does the same to her present, and declares that hers is also 60” around.

Being smart children, however, they know that there two other ways to wrap a tape measure around their boxes to measure the perimeters.

Jessica measures her gift in the second plane. She proclaims that it is 80” around the second way. Zhenya also measures hers in the second plane. She also announces that hers, too, is 80” around that way also. So far a total match.

Finally, Jessica measures hers in the third plane, and declares her perimeter length to be 120”. Zhenya measures hers in the last plane and is disappointed to find out it is only 100”.

Both children agree that it’s the volume of the present that is the most important thing. Jessica claims she has the present with the largest volume. After all, their presents share measurements of perimeter in two planes, and hers is the larger perimeter in the third plane. Bigger is better! Is she correct?

Who has the present with the largest volume? Is it Jessica, is it Zhenya, or do both presents have the same volume?

First, let’s work out the dimensions of the two boxes. We’ll use the x,y,z variables to be the box dimension in those respective axes.

The three perimeters are therefore: 2(x+y), 2(x+z), 2(y+z)

Adding all these three perimeters together gives 4x + 4y + 4z which we can divide by two to give 2x + 2y + 2z, from this we can subtract each individual perimeter in turn e.g. 2(x+y), to give us the result of twice the dimension in the final axis e.g. 2z.

Zhenya’s present perimeters are 60, 80, 100, which sum up to 240. Half of this is 120, so the dimensions of her present are: x=10, y=20, z=30.

Jessica’s present permitters are 60, 80, 120, which sum up to 260. Half of this is 130. Her present dimensions are x=5, y=25, z=35.

The volume of each present is the product of these three dimensions. Zhenya’s volume is 10×20×30 = 6,000 cubic inches. Jessicas’s volume is 5×25×35 = 4,375 cubic inches. Zhenya’s present has over 37% more volume!

This might seem counterintuitive at first, until you look closer at the dimensions of each box. Jessica’s present is flat and thin, but Zhenya’s is closer to a cube in shape. A cube is the most efficient flat sided cuboid container you can make to store volume; other things equal, the present that has dimensions most similar to a cube is going to have the highest volume.

If you are interesting in cylinders instead, here is an article that talks about the optimal dimensions for a can.