Introduction
For this measurement project, my group members (Matthew Evans and Travis Baron) and I chose to measure the area and volume of the Pentagon, which is the headquarters of the United States Department of Defense. It is located in Arlington County, Virginia just across the Potomac River from Washington D.C. To find the volume of the Pentagon we first had to find the area of its pentagonal face, which proved to be a relatively simple task, considering that it was a regular pentagon we had to contend with. In order to accomplish this task, we needed to know the side lengths of the outer walls of the Pentagon as well as its height. These measurements were found with the help of Google and are 921 feet and 71 feet respectively. After finding the area of the face, we had to multiply it by the structure's height in order to find its total volume. The next step in this project was to find the volume of the Pentagon's central courtyard and subtract its volume from the total volume of the structure. Thus, we achieved the final volume of the Pentagon.
Math Calculations and Results
In order to find the area of the pentagonal base, we split it up into three isosceles triangles (two on the sides, one in the middle), two of which were congruent. After this, we found the angle measures for all three of them and dropped a line perpendicular to the base from each, thus making six right triangles. I started by calculating the areas of said right triangles, starting with the isosceles triangle on the far right. The hypotenuse was 921 feet, which leaves us to find the other two legs of the triangle. Since the measure of the angle adjacent to the hypotenuse was 36 degrees, we used the trigonometric function 921 cos(36) in order to find the length of the side adjacent to the hypotenuse, which we found to be 745.1046518 feet. Subsequent to this, we used the Pythagorean Theorem to find the side length opposite to the hypotenuse, which is 541.3502174 feet. Now, all that was left to do was to find the area of the right triangle, which we found to be 201,681.2826 ft². We then multiplied this area by two to find that of the entire isosceles triangle, which is 403,362.5652ft², after which we found the total area of both congruent isosceles triangles, which is now 806,725.1305ft².
Reflection
In finding the volume of the Pentagon, we didn't encounter too many challenges, except for the fact that doing the calculations was a very drawn-out process as we checked them five times to ensure accuracy. Sometimes in the middle of calculation, one of us would forget to solve a formula or other necessary equation and would have to start all over again. It was very frustrating at times to manage all the work, especially since sometimes one person was doing more work than others because communication between group members wasn't the best it could be. Next time when working on a group project, I will do my utmost to communicate to my other group members what needs to be done so something like this doesn't happen again. The Habits of Mathematician we used were Conjecture and Test, Stay Organized, and Be Confident, Patient, and Persistent. Conjecture and Test was used while checking for the right answer multiple times. Stay Organized was used when we had to keep track of the numbers we were using and the multiple attempts we tested. A few other Habits of a Mathematician we applied in this project were Be Confident, Patient, and Persistent because we had to keep our cool while looking for the answer, which took more than one attempt.