- How big is the Bermuda Triangle?
Question(s) To Ask
- What is a guess that is too low?
- What is a guess that is too high?
- What is your best guess?
- What assumptions are we making?
- How can we determine whether the GPS coordinate corresponds with the x or y coordinate?
- What have we learned that may be useful in determining the triangle’s area?
Here are my assumptions below to consider when using this problem:
- The Bermuda Triangle is a triangle whose vertices are not well defined but are generally locations in Florida, Puerto Rico, and Bermuda.
- I am using the latitude and longitude coordinates as my (x,y) coordinates. However, the first coordinate is the latitude which is the y-value and the second coordinate is the longitude which is the x-value. If you want students to figure this out on their own, I recommend giving them a piece of graph paper and asking them to plot and label the points. Then compare it to a map and facilitate a conversation around why they look different. So, as an ordered pair, the coordinates are:
- Miami: (-80.226529, 25.789106)
- San Juan: (-66.1057427, 18.4663188)
- Hamilton: (-64.781380, 32.294887)
- I am being intentionally vague about what “big” means in the question. It should provide an opportunity to attend to precision so you can discuss what we are measuring and what units we will use to measure it with.
- According to Wikipedia (screenshot included in “What You’ll Need”) “the total area var[ies] from 500,000 to 1.5 million square miles” depending on the boundaries and vertices of the triangle. You will notice in the movie clip that in addition to the triangle drawn, the red area is much larger as I assume that shows area roughly in the Bermuda Triangle.
- I am ignoring the fact that the Earth is an ellipsoid and treating the section of the map like it is a flat plane. This is important when converting from degrees to a linear unit like miles. National Atlas (screenshot included in “What You’ll Need”) explains this in more detail. For now I am assuming that one degree equals 69 miles in every direction. Later we can discuss it as a potential source of calculation error.
Here are several different methods students may use:
- While I am intending for this to be a high school level lesson, if you give students a map with scale like the one below you can get a reasonably accurate measurement by using the traditional triangle area formula of 1/2*base*height. I got 470,080 square miles when I used this method.
- Students may choose to enclose the triangle in a rectangle and subtract out the three extra triangles from the total. The way I approached it, I translated the xy-coodinates for Miami to (0,0). That made the xy-coordinates for San Juan (14.12079, -7.32279) and Hamilton (15.44515, 6.505781). I then graphed it on Desmos. The dashed rectangle that inscribes the triangle has an area of 213.5843 square degrees. The triangles on the right, top, and bottom have areas of 9.15702, 50.24138, and 51.70176 square degrees, respectively. That leaves an area of 102.4841 square degrees for the Bermuda Triangle. Based on our assumption of a degree being 69 miles, a square degree would be 4761 square miles. That would give an area of 487,926.8 square miles.
- Students may use the formula for finding the area of a triangle by setting up the coordinates as a 3×3 matrix and finding the absolute value of half the determinant. That also gives the Bermuda Triangle an area of 102.4841 square degrees which is 487,926.8 square miles.
Once students have their guesses, you can play the answer for them.
What You'll Need
- Wikipedia on the Bermuda Triangle’s area and vertices:
- CCSS G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
To begin the class, ask students the following questions:
- How can you classify triangles according to their angles? [right, acute, obtuse]
- How can you classify triangles according to their sides? [equilateral, isosceles, scalene]
- What do you know about the Bermuda Triangle?
1. After the introduction, allow the students to watch the video. Use the handout? What you know about the Bermuda Triangle?
2. You can assign groups for this activity to allow the students to turn and talk and then answer questions. Make sure students record answers on the worksheet.
1. After a brief discussion, distribute the Bermuda Triangle Sides and Angles activity sheet to all students. Answer any questions that students have regarding the activity.
2. Once all questions have been answered and students are ready, allow them to work for 3-5 minutes individually to measure and classify angles. In addition, students need to measure the length of the sides. (You will not need much time for this. During the discussion about types of triangles, prompt students to think about how sides and angles are related.
3. For the next 3‑5 minutes, allow students to share their thoughts with a partner. During these discussions, students will often realize any errors that they made.