My group chose to measure the area and volume of the Pentagon. In order to find its volume, we first had to find the area of the base of the Pentagon. This was not too difficult because the Pentagon is a regular polygon. To find the volume of the pentagon, we first needed to find its side lengths and height (in feet), which were readily available on the internet. Next, we found the height of the whole prism, which was 71 feet according to Google. Then, we decided to find the volume of the central courtyard as well. Lastly, we subtracted its volume from the total volume of the Pentagon itself. We separated the pentagonal face into 6 separate triangles to simplify our way to the answer. We also used the cosine function to find the sides of the right triangles. To use this function, we had to know the angles of the right triangle, so we used a method of finding the angles of a regular polygon. To obtain the most accurate results for the area, we double, triple, and quadruple checked the calculations. Even though this process was long, it provided us more accurate results. In finding the volume of the whole pentagonal prism, we simply took the area of the base and multiplied it by the height of the prism. The last thing left to do was find the volume of the Pentagon’s central courtyard and subtract its volume from the total volume of the Pentagon itself to get our answer.
Math Calculations
We first need to find the area of the pentagonal base of the structure before calculating its volume. To do this, we first found all the interior angles of the pentagonal face. These were easily able to calculate, and we did it by dividing the number of degrees of arc in a circle (360°) by the number of sides on the polygon (5), and then subtracting that number (72°) from the total number of degrees in a triangle, which is 180°. Each interior angle was 108°. We then took apart the pentagon into three separate isosceles triangles and divided those triangles into six right triangles. Thanks to the internet, we were able to find the outer side lengths and the height of the pentagonal prism, which were 921 feet long and 71 feet high respectively. To start, we decided to find the area of one of the right triangles, specifically the isosceles triangle on the right. We already knew its hypotenuse, so what we had to do was find the lengths of the other sides of the triangle. In order to do so, we used the cosine function 921 cos(36) to find the length of the side adjacent to the hypotenuse. The angle 36° is the measure of the angle adjacent to the hypotenuse. We found 745.1046518 feet as the side length of the adjacent side of the triangle. Using the Pythagorean Theorem, we found that the side length opposite to the hypotenuse is 541.3502174 feet. Next, we found its area, the equation for which is A=1/2bh. The resulting area of this triangle is 201,681.2826 ft², which we then multiplied by two, obtaining the area of the entire isosceles triangle, which is 403,362.5652ft². In order to find the total area of both of the isosceles triangles, we multiplied the total area of one of the isosceles triangles by two, which gave us the answer 806,725.1305ft². This is the total area of both of the isosceles triangles. We now have to find the area of the central isosceles triangle. Since we found the adjacent side length of one of the other right triangles (745.1046518 feet), we simply doubled it in order to find the hypotenuse of one of the right triangles of the central isosceles triangle. This turned out to be 1,490.209304 feet. From here, we found the length of the right triangle’s base, which was 460.5 feet long (incidentally, exactly half of the pentagon’s side length).Now, we use the Pythagorean Theorem to find the side length opposite to the hypotenuse of the right triangle, which was 1,029.709304 feet. After this, we found the area of the right triangle which happens to be 237,090.5673 ft². In order to find the area of the isosceles triangle as a whole, we multiplied the area of the right triangle by two, which gave us a total area of 474,181.1344 ft². To find the area of the entire pentagonal face, we added the total areas of all the isosceles triangles together, which gave us a total area of 1,280,906.265 ft². We now need to find the total volume of the Pentagon (excluding the central courtyard). We do this by multiplying the area of the base by the height of the prism, which, is 71 feet. We got the answer of 90,944,344.81 ft³ as the total volume of the prism. The next task was to find the area and volume of the Pentagon’s central courtyard. With the assistance of Google, we found that the area of the central courtyard is 217,800 ft², so all we had to do from here was multiply that area by the height of the Pentagon. This gave a volume of 15,463,800 ft³ for the central courtyard. Then, to find the final volume of the Pentagon, we subtracted the volume of the central courtyard from the total volume, which came to be 75,480,544.81 ft³
Reflection
We didn’t encounter too many challenges but the few that we did encounter, we overcame fairly quickly. The main challenges of this project were figuring out the volume of the Pentagon excluding that of the central courtyard. For the graphic organizer, Rachna and Matthew worked on that. The calculations were mainly worked on by Rachna and were reviewed by Matthew and Rachna. They were checked 4 different times for a more accurate and definitive answer. The slides of this presentation have been worked on by Matthew and Rachna. 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. If we were to do this differently, we would have better communication between group members to make sure that everyone knows the math we are working with and how they can help.