June 14, 2026. A first-principles Fermi analysis of goal-scoring sports.

Introduction

I’ve long been fascinated by the question of why different goal-based sports have a different number of characteristic goals per game. Given that World Cup 2026 is upon us (I just watched Australia win in person!), it seems like a good time to give the problem more thought.

Basics

There are a few factors relevant to number of goals per game:

• $N$: number of on-field players;
• $F$: number of forwards;
• $V$: average speed of ball per game;
• $A$: area of field;
• $T$: total length of game;
• $W$: width of goal.

A very crude model is to think of the field as divided into $N$ regions, $a = A/N$, modelling each player’s reach on average. Kick frequency $f$ per player is given by dimensional analysis Since it is per player, it shouldn’t depend on $N$. as

\[f \propto \frac{V}{\sqrt{A}}\]

up to an $\mathcal{O}(1)$ constant $C$. Hence, the total number of kicks per game is

\[N_\text{kick} = NTf \propto \frac{NTV}{\sqrt{A}}.\]

How often do these kicks go into the goal? Well, forwards exist to score goals, and assuming the ball sits with a player at random, and shots on goal are proportional to area of goal vs length of field (to account for passes, inaccuracy, etc) we get

\[P_\text{shot} \propto \frac{F}{N} \cdot \frac{W}{\sqrt{A}}.\]

Thus, the total shots on goal is

\[N_\text{shots} = P_\text{shot}N_\text{kick} = \frac{FWTV}{A}.\]

Let’s plug this in and see what we get for football (aka soccer in Australia). Our data:

• $N = 20$;
• $F = 3$ (typically);
• $V \approx 1 \text{ m/s}$ (estimate);
• $A = 105 \text{ m} \times 68 \text{ m} = 7140 \text{ m}^2$;
• $T = 90 \text{ min} = 5400 \text{ s}$;
• $W = 7.32 \text{ m}$.

The only figure here requiring some explanation is $v$, which I compute as a geometric mean of a slow pass ($V \approx 0.05 \text{ m/s}$) and a fast pass or shot ($V \approx 25 \text{ m/s}$), $\sqrt{0.05 \times 25} \approx 1.1$. Plugging in numbers, we get

\[N_\text{kick} \sim 1300,\]

or a kick every $4$ seconds, which seems reasonable. The number of shots is

\[N_\text{shots} \sim 17,\]

which is also reasonable. Both numbers are a tad high; looking at data suggests we take the constant of proportionality appearing in $f$ as $C \approx 0.65$ to get $\sim 850$ kicks and $\sim 11$ shots on goal.

Goal-keeping

The real subtlety in our analysis, as in the game itself, is to convert shots on goal to goals. A goalkeeper has an area, but it is a highly mobile one! This requires a little input from physiology. A goalie can cover a (linear, for simplicity) area that depends on three things:

• $w$: wingspan;
• $\tau$: reaction time;
• $v$: speed of dive.

If a shot is launched at distance $D \sim \sqrt{a}$ in a forward zone, and with speed $V’$, it takes time $D/V’$ to arrive and the goalie can move a distance

\[d = v\left(\frac{D}{V'} - \tau\right)\]

assuming they have enough time to react at all. This gives them an effective width

\[w' = w + 2 v\left(\frac{D}{V'} - \tau\right) = w + 2 v\left(\frac{\sqrt{a}}{V'} - \tau\right).\]

So we model the probability of a goal as simply

\[P_\text{goal} = \frac{W - w'}{W} = 1 - \frac{w}{W} - \frac{2 v}{W}\left(\frac{\sqrt{a}}{V'} - \tau\right).\]

Again, let’s plug in the numbers and see if we get a reasonable answer. We use

• $w = 2 \text{ m}$;
• $\tau = 0.2 \text{ s}$;
• $v = 3 \text{ m/s}$;
• $V’ = 25 \text{ m/s}$.

If we plug all of these in, we get

\[P_\text{goal} \approx 0.27 \quad \Longrightarrow \quad N_\text{goals} = N_\text{shots} P_\text{goal} \approx 4.6.\]

Since $N_\text{shots}$ has a constant of proportionality $C \approx 0.65$, this reduces to $3$ goals, not far from the statistical average of $2.5$. This is arguably closer than we had any right to get!

Hockey

Let’s see if this works for another sport. In hockey, we have the following:

• $N = 12$;
• $F = 3$;
• $V = 2.5 \text{ m/s}$;
• $A = 61 \text{ m} \times 26 \text{ m} = 1586 \text{ m}^2$;
• $T = 60 \text{ min} = 3600 \text{ s}$;
• $W = 1.83 \text{ m}$.

Putting in numbers in, we obtain

\[N_\text{passes} = \frac{NTV}{\sqrt{A}} \approx \text{2700},\]

which is too large by a factor of $3$ or so. Similarly,

\[N_\text{shots} = \frac{FWTV}{A} \approx 31,\]

which is too small by a factor of $2$. So there is some tension. Finally, if we set

• $w = W/2$; This is contentious! Clearly we can’t have wingspan $w > W$, so we need to adjust the wingspan. Unlike football, where the goal is much wider than it is long, in hockey the aspect ratio is closer to a square, and areally, a goalie takes up about half the goal. So we project the same proportion to 1D.
• $\tau = 0.2\text{ s}$;
• $v = 3 \text{ m/s}$;
• $V’ = 45 \text{ m/s}$

then computing gives

\[N_\text{goals} = \left[1 - \frac{w}{W} - \frac{2v}{W}\left(\frac{\sqrt{a}}{V'} - \tau\right)\right] N_\text{shots} \approx 10.\]

Applying the same constant $C \approx 0.65$ gives a total of $6.4$ goals, which is shockingly close to the average.

Exercise. (a) The interested reader should apply the dimensional analysis to AFL with $C \approx 0.65$. You should find good agreement with the statistical average of $26.5$ goals! (b) Check the model for basketball and explain why it breaks down.

Conclusion

Our equations are crude but work far better than they should, particularly since we neglected possession (though our low ball speed $V$ probably partially accounts for this). I suspect this means either (a) we massaged the numbers to get what we wanted, or (b) goal-based games are kinetically straightforward enough to capture bulk statistics easily. Or a bit of both! Still, a fun way to bask in the glory of an unexpected win.