Question: How many footballs could you fit into Wembley Stadium?

You (interviewee): 

To start off with, I will make the following assumptions about Wembley Stadium:

1. The stadium is set up for a football match (as opposed to a concert, in which case the stage and props might take up additional space, thus reducing the number of footballs that could fit into the stadium) and there are no people in the stadium; and
2. The stadium is a cuboid-like structure (i.e. any rounded/sloping structural edges will be ignored). This assumption will make it easier to perform the necessary calculations to estimate its volume.

I will now make the following estimations in order to calculate the approximate volume of Wembley Stadium:

1. The football pitch is approximately 100 metres long and 70 meters wide and there is around 10 metres of space between the pitch and the seats;
2. There are approximately 80 rows of seats in each stand of the stadium. Each row is approximately 1 metre deep and adds approximately ½ metre of height. The top row of seats is 10 metres below the roof. This makes the height of the stadium approximately 50 metres;
3. The spaces between the pitch and the seats, the non-seating areas (e.g. gangways) and the perimeter of the stadium (e.g. the bars, restaurants and offices etc.) add approximately 20 metres of depth to each side of the stadium.

This gives us the following estimated dimensions to work with:

1. Stadium length of 300 metres: 100m (pitch) + 160m (seats either side of the pitch) + 40m (extras either side of the pitch) = 300m
2. Stadium width of 270 metres: 70m (pitch) + 160m (seats either side of the pitch) + 40m (extras either side of the pitch) = 270 metres
3. Stadium height of 50 metres: 40m (80 seats x 0.5m height per seat) + 10m (space above seats) = 50 metres
4. Stadium volume of 4,050,000 cubic metres: 300m x 270m x 50m

I will initially treat footballs as if they are cubes, with the same diameter as an average football. This will allow me to make a direct division of volumes to obtain the answer (the estimated volume of Wembley Stadium divided by the estimated volume of a football).

Based on the assumption that the diameter of an average football is 0.2 metres, the volume of a single “football” would be 0.008 metres³ (0.2 x 0.2 x 0.2). 

This suggests that the number of footballs that could fit inside a completely empty Wembley Stadium is: 4,050,000/0.008 = 506,250,000 footballs. 

However, this does not take into account the fact that there are numerous seats, internal walls and windows, restaurant/bar fixtures and fittings, ticket offices, windows, pillars etc. inside the stadium, which reduce the space available. In addition, more spherical balls could fit inside the stadium than solid cubes. 

Accordingly, I will make the following estimations and adjustments:

1. The seats take up approximately 5% of the stadium;
2. If I were to estimate the amount of non-space in a much smaller structure with which I am more familiar (for instance, a restaurant or a house) I would say that there is around 10% non-space. Applying this logic to a stadium, I will therefore estimate that the internal walls, pillars, windows, fixtures and fittings take up another 10% of the stadium; and
3. 5% more spherical balls than solid cubes (with identical diameters) could fit inside the stadium (as spherical balls have a smaller volume than cubes with identical diameters).

On the basis of these estimations and adjustments, the figure should be reduced by 10% (5% less space because of seats + 10% less space because of walls, fixtures, fittings etc. - 5% because spherical balls have a smaller volume than cubes). This gives us the estimated conclusion that approximately 455,625,000 footballs could fit inside Wembley Stadium. 

In an interview, you may be given a pen and paper to use. If this is the case, use it! If relevant, draw a picture of the object you are analysing (e.g. a stadium or office building) and/or map out your thought process as a diagram (see the framework below). This will not only help you to remember your estimations, but will also help the interviewer to follow your thought process.