With the World Cup in our hemisphere, and the US squad having started out with a win over Ghana, my thoughts turned to the mathematics of soccer. My friend Henri Picciotto has a nice page about the shooting angle, the angle within which a shot is on goal, so I thought of using Sketchpad to explore this idea.

Use the sketch below to find the locus from which the striker, represented by point S, has a 15-degree shooting angle (plus or minus 10%).

When you drag point S closer to or farther from the goal, what do you notice about the point? How does this help you find the locus?

After experimenting a bit, turn on tracing and try to trace the locus as accurately as you can. Then try a different angle (by clicking and changing the target-angle parameter). What shape does the locus have? How can you explain this shape?

A related challenge is to imagine the striker running down the field, toward the goal line, as in the animation below.

When the striker makes a run from midfield straight toward the goal line, where is her shooting angle the greatest? (It may help to turn off the Run! button and drag the striker by hand.) Once you find this spot, mark it.

To change where she makes her run, drag the dashed red line up or down. Mark the greatest-angle spot for different runs. What pattern do you think you see? How can you explain this pattern?

Just as the fun of soccer is in the playing, the fun of these mathematical soccer challenges is in exploring them, so I provide no solutions here. But if your experimentation gives you some ideas about how to find the location of the maximum shooting angle for a given run, you can download the sketch here and try to construct that point, and even the locus of that point for the striker’s different runs. (If you don’t already have Sketchpad, you can download it here.)