How Much Money Is A Quarter Brick

I can measure the height, length, and width with a normal lengthometer (some might call this a ruler). What about the bumps on top (I believe Lego calls them studs.)? I will ignore them—this isn't the real size anyway. Using these measurements, I can calculate the volume in cubic centimeters.

But there is another volume measurement in units of Lego—the studs. For the brick above, it seems clear that the length and width are two studs long. What about the height? If you take the thinnest Lego piece, which resembles a slice, you find that three of them is the same height as one "normal" brick.

And so I will assign this "normal" brick a height of three slices, although I am sure Lego has specific term for this dimension. To find the volume, I simply multiply length times width times height. However, a height unit isn't the same as a length or width unit, so this might seem meaningless. I think I will stick with the volume measurement in centimeters.

Actual Lego Data

Lego offers many kinds of pieces. I am going to stick with the basic shapes—no specialized pieces. Here's a look at the stuff I measured:

RHETT ALLAIN

With the mass and size data, I can plot mass versus volume:

The data looks fairly linear—well, linear enough for me. The slope of this fit is also a great way to estimate the density of a Lego piece. Slope is defined as the rate that the vertical variable (mass) changes with respect to the horizontal variable (volume). This is just like mass divided by volume—the density. Looking at the slope of this fitting line, the average Lego density is 0.565 g/cm³. This seems plausible. I always like to compare densities of object to that of water which is 1 g/cm³. The Lego density is lower than water so that they would float—assuming that the air stays inside of the piece. Again, this is reasonable.

But why does the data look like it has two linear functions mixed together? Probably because there are two Lego densities. In this study, I only examined single-height blocks or the three-height blocks. The shorter ones offer less empty space—this would yield a higher density than the taller blocks. That's just my guess—you can examine Lego density further as homework.

Price Per Pound

How do I link the Lego density to the price per pound? First, I can determine the price per gram with a simple unit conversion. This would be 7.9 cents per gram. Now using the 10.4 cents per piece, I can do the following unit trick:

La te xi t 1

If I invert that, I get 1.35 grams per piece. This could be interpreted as the average mass of a piece in a kit—if the $8.99 per quarter pound is realistic. What would a 1.35 gram Lego piece look like? Based on my data it would fall somewhere between the tall 2x2 and the tall 4x1 brick.

How about just one homework question. Find a ratio of Lego pieces such that the average mass is exactly 1.35 grams. You should be able to see my mass data if you click on the plotly graph above. You will need that or you will need to determine your own Lego mass data.