The Lacrosse Goal
My experiences and interests as a lacrosse player have driven me to complete a project concerning the lacrosse goal. As one may already know, the main purpose of the game of lacrosse is to score a goal against the opponent. In order to win, a team needs to obtain the most goals. With this importance of scoring and my fascination with lacrosse in mind, I will take it another step using geometric principles. Although it may not be thought of as essential to winning, the area in which a player may shoot at does indeed have a significant impact on the probability they will score.
By observing various angles from which a player can shoot, several conclusions can be made. For example, as the lacrosse goal is viewed from the side, it is apparent that the standard six feet by six feet goal does not apply. Only if it is seen as head-on does the goal appear as a square. However, from the side, it takes the shape of an isosceles trapezoid. The area of this trapezoid constitutes the apparent area a player may shoot at looking from a particular angle, rather than the actual area of the goal36 square feet. The pole in the foreground remains an invariable six feet while the pole in the background differs. Through utilizing a few assumptions, I was able to calculate the area of the trapezoid. These assumptions include as follows:
An official lacrosse goal was used measuring 6 feet by 6 feet from the middle of the poles, but for our purposes, we will take for granted that the inside inch of the pole is included as part of the shooting area.
Secondly, I have used a screwdriver to represent the point from which a player shoots, but the distance away from the goal and the height of the player is irrelevant, for the area of the goal stays the same.
12 feet has been made as the constant length for one side of the angle as a control.
There will be slight errors in the computations, since nothing measured will be perfect.
The method I employed involves trigonometry, proportions, and the area formula for trapezoids. First of all, using string, a screwdriver, a tape measure, and protractor, I have taken pictures of the appearance of the goal at angles in intervals of five degrees while keeping one side twelve feet long. Before anything, since a perpendicular line is the shortest distance between a point and a line, a perpendicular must be drawn from the bottom corner to the opposite line. It is the minimum distance required for the ball to be shot and go in. With the measure of the angle given, trigonometry can be applied with the sine of the angle multiplied by the hypotenuse (always equal to twelve feet) to get the measure of the distance of this perpendicular line, which is then equal to the height of the trapezoid. We then need the measure of the second base of the trapezoid and to acquire this, I used proportions by measuring the length of the segments with a ruler. I compared their ratio to the ratio of the actual lengths, with six feet as the actual length of the larger base. Finally, we use the formula (b1 + b2) h, pertaining to every possible trapezoid if the angle, pole length (6 feet), and side length (12 feet) remain the same. At 30 degrees, the area is a square and the formula s^2 is consequently used. This area is the actual area of the goal.
Now that we have analyzed the apparent areas and actual area of the goal at different angles, we still havent factored in the fact that a goalie is present to block the shots taken. The area available for shooting will be significantly decreased based upon the size of the goalie and the goalie stick is allowed to have a head 10-12 inches wide compared to 6.5-10 inches. Even with all these drawbacks for the shooter, a ball surprisingly as small as 7.75 to 8 inches in circumference, made of solid rubber, and weighing 5 to 5.25 ounces can sure be fleeting. In conclusion, as the angle increases on both sides of the goal, the area available for shooting also increases, and vice versa. The apparent area of the goal is actually a trapezoid and always less than the actual area of the goal, which is 36 square feet. Through using proportions to find the lengths of the bases, trigonometry for the length of the height of the trapezoid, and the formula for the area of the trapezoid, each apparent area of the goal (depending on the angle at which it is seen) can be calculated. Thus, with this knowledge, more scoring is converted into more winning.