Where Should You Shoot? Uncovering the Optimal Soccer Goal Positions with Geometry

Soccer is a popular sport, and players often wonder which positions on the field give the highest shooting success rate.

Assuming (1) the ball is a point mass, (2) its trajectory is parallel to the ground, and (3) there is no defending opponent, we analyze the problem.

The standard football field dimensions are shown below.

Using plane geometry, we can find a point on the edge line that maximizes the shooting success rate; the larger the distance from this point, the higher the rate. This point is defined as the optimal shooting point. The field is divided into three strip regions, and a rectangular coordinate system is established with the goal line as the x‑axis and the perpendicular bisector as the y‑axis.

For a point chosen inside region A, if the distance to the goal line is kept constant, the moving point can only slide along a line segment. By geometric relations, the product of certain distances remains constant, leading to an equation of an equilateral hyperbola that represents the locus of optimal shooting points in that region.

Similarly, the optimal‑point locus for region B is derived, and for region C the analysis shows that the closer the point is to the central axis, the higher the shooting success rate, contradicting the common belief that being nearer to the goal is always better.

Further discussion introduces the concept of shooting‑equivalent lines: arcs on which every point yields the same shooting effectiveness. These arcs are concentric circles centered on the goal line, and their equations are given in terms of a parameter.

The model is coarse because of the simplifying assumptions and ignores many real‑world factors, but it provides a geometric framework for understanding optimal shooting positions on a soccer field.