June 14, 2026. A first-principles 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 already, we have some tension. Finally, if we set

• $w = 2\text{ m}$;
• $\tau = 0.2\text{ s}$;
• $v = 3 \text{ m/s}$;
• $V’ = 45 \text{ m/s}$

then computing gives

\[N_\text{goals} = \left[1 - \frac{w}{W} - \frac{2v}{W}\left(\frac{\sqrt{a}}{V'} - \tau\right)\right] N_\text{shots} < 0,\]

which is nonsense. This means that, in theory, hockey goalies can always block shots, so what have we missed? Lots of things, since we did a Fermi estimate, but the interesting part here is tactics. These include the following strategems:

• Screens. Place a player in front of the goalie to obscure the play, changing reaction time.
• Deflections. A shot bouncing off something creating a new shot, new reaction time.
• Passing shot. A near-goal pass just before the shot on goal fakes out the goalie.
• Approach. Taking the shot at point-blank range makes reaction time irrelevant.

So the breakdown of our dimensional analysis reveals something cool: hockey works in a fundamentally different way from soccer.

Conclusion

Our equations for football work beautifully. For hockey, they fail miserably. This is because football is based on passing, possession, and shot-taking with reasonably simple boundary conditions. In contrast, the relative size of goal and goalie in hockey force a variety of new strategems which, heck, make the game so darn fun, albeit for different reasons from football.

Exercise. The interested reader should apply the dimensional analysis to AFL with $C \approx 0.65$. You should find good agreement with the statistical average $N_\text{goals} \approx 26.5$ goals!