Why triangles allow more room than squares


After reading last week’s column, Michael Wang’ombe asked me to help him understand how a large hall can accommodate so many more pupils in a triangular grid than in a square one. I will not go over the detailed geometry as that was explained in the last article.

Last week we saw that a classroom measuring 18ft by 24ft can fit 35 students in a square grid and 36 in a triangular pattern. That was just one extra student.

Suppose it was a hall measuring 36ft by 48ft: how many candidates would fit? We will use the same limitations we had last week — a minimum of 4ft between candidates and one foot from the wall.

In a square grid arrangement, we get nine pupils along the 36ft width and 12 along the 48ft length. This makes a total of 108 students.

For the triangular pattern, we get 14 rows; seven with 9 students and the other seven with 8. The total comes to 119. That is, we fit an additional 11 pupils or about 10pc more.


Now, the triangular pattern has a hidden advantage: the back-to-front distance is a lot more. For candidates in a square grid, the distance from one student to another, in front or behind, is 4ft.

In the triangular pattern, the side-to-side separation is also 4ft but the back-to-front separation is almost 7ft (6.93ft)! The reason for this is the staggered arrangement of the rows. I believe this makes it more difficult to cheat in the exam.

Perhaps the question that Wang’ombe did not ask is why the triangles fit more students than the squares. There are two ways to explain this. First is the number of nearest neighbours a candidate has in each pattern. (By: the way “nearest neighbour” is a proper technical term in the science of crystallography)

In both patterns, the nearest neighbour is 4ft away. In the square pattern, a student in the middle part of the hall will have four nearest neighbours. What about in the triangular arrangement? One might expect that a similarly positioned pupil in the triangular arrangement would have three nearest neighbours — after all, a triangle has three corners.

That is not the case at all! If you sketch the triangular grid, you will find that a pupil in the middle part has six nearest neighbours. Therefore, this pattern packs the candidates more tightly even though they are still 4ft apart.

The second way of understanding the phenomenon is to look at the area reserved for each student. A 4ft-by-4ft square has 16 square feet and four students. That is, each student takes 4sq ft.

The area of a 4ft triangle is about 7sq ft. This will have three pupils — one each at the corners. Therefore, the area per student is 2.3sq ft. Clearly, there is better utilisation of space in this case. And you thought Pythagoras Theorem was a waste of time!