Spreadsheet Math: Triangle Calculator

In high school math, there are really only a few checkpoints that mark one’s right of passage of mathematical prowess. Mastering quadratic functions (e.g. using the quadratic functions) is perhaps the first and most widely recognized. Solving triangles is another.

Part I: Solve for right triangles and trigonometric ratios

Many of us learned that a simple way to solve right triangles (i.e. triangles one being equal to 90°) can be done by employing SOH-CAH-TOA and the Pythagorean Theorem. Although we may recall solving hundreds of these types of problems, all of them can be categorized into one of four unique cases¹:

• given two legs, solve for everything else
• given the hypotenuse and one leg, solve for everything else
• given the hypotenuse and an angle, solve for everything else
• given one leg and an angle, solve for everything else

Although the inputs (numerical values assigned to each quantity) may change from problem to problem, the method for obtaining the solution does not. Those who do not understand this concept do not truly understand math. Math is not the study of becoming a calculator, but rather the study of patterns that allow calculators to work. That is why to test my students’ understanding of the mathematical concept, I have them build calculators. This assignment is no different.

A template for the four cases to be solved.

To get things started, I have my students make a copy of the above template. Students then work in teams to figure out what the formula needs to be in order to make it work. Some savvy students will figure that spreadsheets probably has some trigonometric functions worked into it, and they are correct. However, simply using =sin(C7) or =tan(H17) won’t quite work (more on why later), which is why they should be testing it with known triangles and/or checking against their store-bought calculator.

Simply enough, once they work out the trigonometric ratios for the first case (upper left), they can effectively copy and paste into the spaces provided for the second case (upper right) since the spreadsheet is smart enough to update the reference cells accordingly. The third (bottom left) and fourth (bottom right) cases require a little more ingenuity since two sides are required to solve for the trig functions, but only one is provided initially.

It is at this point that students will be forced to try the trigonometric function commands if they hadn’t already. It is also at this point that they will “discover” there is a problem. As an instructor, we have a couple options here (depending on what your ultimate goals are):

1. Preempt students with the solution that they should input a corrective (conversion) factor of *pi()/180; or
2. Pretend you don’t know why, allow them to struggle and encourage them to seek out a solution from the internet

Various resources available to explain and help students solve the trig function issue in their calculators.

Part II: Solve for non-right triangles

After mastering SOH-CAH-TOA and the Pythagorean Theorem comes the test of solving triangles that do not involve right angles. To do this, students have to use the Law of Sines, Law of Cosines, Triangle Sum Theorem, and/or some combination thereof. Although there are several scenarios that can occur, I limit this activity to the following four:

• Use Law of Sines to solve for a triangle with known AAS values
• Use Law of Sines to solve for a triangle with known ASA values
• Use Law of Cosines to solve for a triangle with known SSS values
• Use Law of Cosines to solve for a triangle with known SAS values

For solving these problems, it is particularly helpful if students draw the following diagrams to establish the relationship between the different sides or angles and the respective expression needed to obtain their values.

Cases to be solved using Law of Sines

Cases to be solved using Law of Cosines

Although it is an interesting problem, I find it best to avoid scenarios that could yield the ambiguous case for Law of Sines. Even though it could be easily rendered a non-issue by using the Law of Cosines instead, I believe it is an interesting problem that serves better for “next level” problem solving.

Again, I provide a template for the groups to fill out, as the primary focus of this assignment is to understand how to apply and manipulate the various trigonometric laws to solve triangles for a given scenario. The secondary goal is for them to realize that given a particular scenario, the only thing that really changes is the input/outputs and not the formulas themselves. This means once they get their formulas right, they will have effectively solved every problem of that type that they are ever to be given.

The template provided for solving non-right triangles.

Part III: Testing the Calculator

The last part of this project is assessing

1. whether or not the calculator works, and
2. whether or not the student actually understands how to use it

Note that my goal here is NOT to assess their ability to discern which calculator applies to a particular triangle problem. Although you could decide to do that if you so choose, I save that for the actual pencil and paper test at the end of the chapter.

To do this, I have each group open their calculator on one computer and then gather in the back of the room with the rest of the class while I hand out a paper with test values next to their computer. Each student within the group is responsible for obtaining two sets of values from their calculator. During their turn, each student has 1 minute to enter the values provided, and copy the outputs onto their group’s paper. A sample of what the answer sheet might look like can be seen below.

After the “test” is completed, I have the groups turn in their answer sheet and I compare them to my own and grade it as one might a regular test: total number of matching correct answers divided by number of expected number of answers is their score (I always make these kinds of projects worth 100 points).

Because I designed this calculator ahead of time to ensure it works, I can also use it to customize the “test” by changing input values without having to work out a new answer key. For a math teacher, this is invaluable and is what has made Kuta Software so popular. Below is the solutions to the above sample problems.

And that’s it. Through this project students are

• getting more exposure to how to use spreadsheets to neatly organize and process information
• learning somewhat advanced formulas (or, so they will seem for at least for the time being)
• having to confront expectations not being met and figuring how to resolve those issues (i.e. using the internet as a tool to find out how to rectify the fact that spreadsheets do angular computations in radians rather than degrees, inverse trig functions that they need are not arcsin() or sin-1() as they likely had anticipated)
• using new approaches with technology to solve traditional problems

I hope you enjoyed and feel free to share with others or ask questions in the comment section below.

¹ There are of course also the cases of solving a triangle given a trigonometric expression, but that is essentially a subset of being given any two sides and solving everything else.