Sports players and enthusiasts often consider the angle at which the object should released to ensure the best outcome. These discussions and considerations, whether they include discussions about parabolic arcs, triangles or circles, all rely on angle measurement.
The trajectory of a ball during a soccer game varies greatly depending upon its position at any given time (Linthorne & Patel, 2011). The soccer ball could be positioned on the ground, attain an altitude of 50 or more yards from a drop kick, thrown into a game with projectile motion where it reaches a maximum height of 3 yards, or juggled inches from the ground as it is in motion. In addition, the vertical and horizontal movement of a soccer player is another variable to consider when evaluating the point from which a ball should be released.
As I thought deeply about this situation, I decided to explore the optimum point at which a soccer player should shoot a ball as she is running along a sideline. Because soccer fields are in the shape of a rectangle, I constructed a diagram in Geogebra to represent the sideline and goal line.
I started by creating line segment AB, with a length of 10 units. Line segment AB represents the sideline along which the player is running to shoot the goal. I then constructed the goal line as a line perpendicular to line segment AB, or the side of the soccer field which would containing the goal post. Once the goal line was constructed, I created a line segment CB, with a length of 10 units, along this line. I also added line segment EF, giving it a length of 1.75 units, to represent the goal post.
Point H symbolizes the position of the soccer player running down the sideline, while the “red zone”, or the area within triangle EHF, represents the optimum zone within which the soccer ball could travel towards the goal post. Using a circle to connect Points H, E and F allowed me to see angle EHF as an inscribed angle within the circle. I know the measure of an inscribed angle is equal to one-half the measure of the central angle that subtends the same arc.
I then used an oscillating animation to show the movement of Point H along the sideline and observed both the central and inscribed angles. The central angle reached a maximum of 18.11 degrees, while the inscribed angle, EHF, reached a maximum of approximately 9.05 degrees. Using these estimated values, I determined the soccer player will have the best shot at 9.05 degree angle.
Interestingly, the optimum shooting location along the sideline is also the point at which the circle becomes tangent to line segment AB, which, along with determining the radius of the circle I created, could help me to calculate the soccer player’s displacement from the goal post in two directions.
The maximum point at which the circle becomes tangent to line segment AB is also the optimum shooting location, or the point where the angle is the largest. Using the measurement tools, I determined this is the point where the player is approximately 5.5 feet from the intersection of the sideline and the goal line.
Constructing a model of a soccer field
After this construction, I began to think more about how the various measurements within my original of my diagram impacted the optimum point from which the soccer player should kick the ball to score a goal. A few variables I considered were the width of the goal post, the distance between the goal post and the sideline, the length of the sideline and the length of the goal line. I also assumed the soccer ball would remain on the ground for each of my subsequent explorations.
The ideal soccer field will have a width of 75 yards and a length of 120 yards, and the goal post has width of 8 yards (“Soccer Field Dimensions, Size, Diagram”, n.d.). I used this process to recreate my diagram using a scale where 1 unit represents 10 yards.
When developing a model of a soccer field, I constructed a complete rectangle to investigate many variations of the problem. With this model, the size of the angle decreases to 6.14 degrees and the optimum shooting location is the point at which the soccer player is 35.8 yards from the goal line. This distance is less than what I expected, based on my earlier diagram, where the optimum shooting location was more than half the distance from the goal line and the other end of the sideline.
Shooting from the halfway line
The halfway line of a soccer field, represented by line segment PQ, runs perpendicular to its sidelines. From this location, if a player were running along the halfway line, the optimum shooting location would be the center of the field. When the player is 37.5 meters from each sideline, the angle measure would be 7.62 degrees.
Shooting from along the penalty arc
The penalty arc is a semicircle with a radius of 12 yards surrounding the goal post. To construct the penalty arc, I created a perpendicular bisector of the goal post, then constructed a circle from the center of the goal post with a radius of 12 yards. Angle MUN represents the angle at which the player takes a shot from penalty arc. Much like shooting from the halfway line, the optimum shooting location would be the center of the penalty arc. At this point, the angle measure would be 36.82 degrees.
Shooting from the center circle
The center circle on a soccer field has a 10 yard radius. When a player is on the center circle, on the side closest to goal post, if she were to position herself at a point along the perpendicular bisector between the goal post this would be the optimum point for shooting a goal. The angle measure is 9.14 degrees.
Running along a diagonal
What might happen if the player runs along a diagonal? Line segment VW represents the diagonal along which she runs.
As the player runs along this diagonal, the angle increases from 4 to 180 degrees the closer the soccer player gets to the goal post.
Connections to Teaching Mathematics
This exploration could be used in a high school geometry classroom. The student expectations from the Alabama Course of Study (Alabama State Department of Education, 2016) I felt most closely aligned to this task were:
G-C2: Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
G-CO-12: Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
G-MG-1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
Possible lesson outline:
- Use dynamic geometry software to allow students to explore the angle at which light enters the human eye. Using the following image as a guide, students could explore relationships between points from which light extends from the focal point through the iris of the eye.
Image taken from https://www.nkcf.org/about-keratoconus/how-the-human-eye-w
- Next, have students identify 3 points (including the focal point) and construct a circle superimposed over the image of the eye to connect the points. Using the measurement tools within the dynamic geometry software, students could make conjectures about the relationship between the angle and the arc of the eye.
- Identify the parts of the constructed circle as the inscribed angle and intercepted arc.
- Introduce the central angle and ask students to make observations about the size of the inscribed angle, the central angle and its intercepted arc within the context of the situation.
- Encourage students to generalize the relationship between inscribed angles, central angles and intercepted arcs. Ask students to consider other situational examples of how the position of the inscribed and central angles may change.
References
Alabama State Department of Education (2016). Revised Alabama Course of Study: Mathematics. Montgomery, AL
Linthorne, N. P., & Patel, D. S. (2011). Optimum projection angle for attaining maximum distance in a soccer punt kick. Journal of sports science & medicine, 10(1), 203.
Soccer Field Dimensions, Size, Diagram. (n.d.) Retrieved from https://www.sportsfeelgoodstories.com/soccer-field-dimensions-size-diagram/
The Chronicles of Algebrea 