# Why are triangles so prevalent in high school geometry?

A colleague and I recently discussed what we call the "Triangle Trap." High school geometry covers a very large unit reflecting the common core:

• Classifying Triangles
• Triangle Angle Properties
• Constructing Triangles
• Properties of Triangles (e.g. Equilateral, isosceles)
• Triangle Similarity
• Triangle Congruence
• Triangle centers
• Pythagorean Theorem
• Special Right Triangles
• Right Triangle Trig
• Law of Sines and cosines
• Area of Triangles

This represents almost a third of our year. And our students get triangle fever -- they get sick and tired of triangles, and honestly so do we. There are so many other exciting topics to delve into.

What is the value of placing this large of an emphasis on triangles in a secondary geometry context?

• To give all students a chance to succeed in university, should they choose that path. Many large American universities assume students are at least familiar with many of those topics, even in most beginning courses. – Chris C Jan 4 '15 at 16:31
• @ChrisC That's just pushing the question one level farther back. Also, many, many large universities are not in the USA. – David Richerby Jan 5 '15 at 8:27
• I am not going to say why they are so prevalent. But I think, if you can relate triangles to their uses, it would be very helpful and interesting for the students and for you too. – Karthick S Jan 6 '15 at 10:57
• Because they represent the middle class in Flatland. – vpipkt Jan 6 '15 at 15:38
• It may be interesting to take Euclid's Elements, the books on plane geometry, and see what portion of those propositions are on triangles. Do you think it would be substantially less than the one-third in the question? – Gerald Edgar May 3 '15 at 13:15

TL;DR: It's not the triangles that are interesting; it's the mathematical concepts that can best be explained by using one of the most primitive geometrical shapes.

The reason for intensive use of triangles goes beyond knowledge about triangles per se.

It's the act of and steps in proving a theorem that's important to learn at this stage - start with some knowledge you "know for sure" (You probably don't want to use the word "axiom" at this stage), formulate logical deductions, arrive at something new you know must be true even if it seems counter-intuitive. This works much better with geometry than with algebra, because high-schoolers don't have ZFC as a foundation to build upon, but you can introduce Euklid's axioms easily, because they're so intuitive.

Also, you can tackle many proofs in a purely geometrical way, or by transforming your geometrical representation into a coordinate system. This is easy for straight lines, but much harder for anything else, so triangles are, again, a good way to introduce the concept. This teaches your students that different fields of mathematics can profit from each other, and in some cases, something that might be hard to prove using geometry alone can be proven much easier in arithmetics. Or vice versa. Which shows to your students, as well, that sometimes there are several wildly different methods of proving something which, at the end, provide the same result.

• +1 Thanks to TL;DR the idea can be grasped in seconds. Most higher voted answers have similar content quality but don't use formatting to their advantage. – Legat Jan 6 '15 at 13:07

This is not an answer, but I share what I take to be your skepticism. I think polygons are a rich topic. For example, just understanding that every simple polygon can be triangulated would be an achievement. And it would lead to the often surprising conclusion that all $n$-gons, regardless of shape, have the same sum of internal angles: $(n-2)\cdot180^\circ$. Another surprise: Not every simple polyhedron can be tetrahedralized.

I see one important reason: triangles are simply the simplest non-trivial configurations of points.

One point configuration are boring: every two points are mapped one to the other by a translation. In other words, every point can be translated to the origin, which thus represents very well every single point.

Two-points configuration are only slightly more interesting: any pair of points can be moved by a translation and a rotation (thus preserving all geometric properties) to a pair $((0,0), (x,0))$ with $x>0$ being the distance between the two points. So the distance is the only meaningful geometric quantity for a pair of points. Now, if one allow for dilation, which preserve all dimensionless geometric quantities (e.g. ratios of lengths), then there are only two type of pairs: twice the same point, and two different points. Boring.

Three point configuration are incredibly rich: there exist a two-dimensional configuration space of pairwise non-similar triangle. Highly non-boring!

Note that the same arguments make triangle interesting in any dimension. As a side note, when one studies affine geometry without a Euclidean structure triangle become boring again. When one studies the projective line, triples of points are boring but quadruples lead to the cross-ratio, etc. So this point of view really make this high-school stuff fit into a much deeper and wider framework.

TL;DR Triangles helped me understand both unit circle and trig functions. Super cool and super useful. Didn't really use much of my other geometry. Don't drop triangles. If you do, you should have a good reason and better replacement.

I am not a mathematics educator. But I did take junior high geometry that covered all those topics. Here's a student's perspective.

I don't see a topic on that list I regret studying. I always thought studying triangles in-depth made it easier to appreciate the role of triangles in defining the trig functions using the unit circle. I took Trigonometry freshman year of high school. If I had just learned SOHCAHTOA alone, I would have never appreciated the important the properties of right triangles versus nonright triangles. For me, my first look at trig was seeing how geometry could give me a new powerful mathematical tool (ie sin cos tan) through the geometry I has learned. Thus, understanding many properties of triangles gave me a better appreciation for how trig functions are defined and why right triangles are useful for that. I thought trigonometry was beautiful in part because I was so familiar with triangles.

That said, I feel my future math courses never used much of the geometry I learned in that course. I have never used SAS or SSS ever since then. In fact, triangles were one of the few things I actually used after that course. Only many years later when I began studying differential geometry and non-Euclidean geometry did I really feel a curiosity to revisit the principles I learned in my geometry class. And I personally always was disappointed by this. Now that I reflect on it, most courses I took in high school I have used. Geometry would be an exception in that much of it I haven't used.

So perhaps if curriculums made more use of non-triangle concepts, it would be worthwhile to study stuff besides triangles. But, especially if you do physics or anything involving Fourier methods, you want to have a good grasp of trig and I feel triangles are a key piece of that. In other words, I wouldn't change how triangles are taught unless I see topics I know will be more useful that should replace them.

I actually would have liked to study stuff besides triangles more in-depth. So I am not necessarily pro-triangle. But I am always trying to invest my time in things that are useful and triangles clearly were the most valuable part of my geometry course.

EDIT 1:

All of the above was, so to speak, in defense of triangles. But suppose there are better things to teach. Like what? If teachers want to teach less about triangles, I would be interested to hear a proposal for what should be taught in place of them. This proposal should keep in mind that secondary school isn't designed to train mathematicians but to provide children and young adults mathematical tools that will be useful for their professions. This ranges from finance to physics and beyond, so we aren't looking for merely the most interesting math but the most useful. So such a proposal should (1) specify what use the math would serve and (2) why this is more useful than studying triangles.

EDIT 2:

I also think you could simply break up the triangles into separate units instead of bundling them together. This would allow you to teach other topics in between. For instance, split off together the trigonometric properties of triangles as well as the laws of sines and cosines. Make those a separate unit and do that unit later in the year. It should be workable I would imagine without having to drastically refresh the students on everything.

• +1, I never would have grasped the unit circle without understanding a right triangle. My geometry teacher made a point to us to show how non-triangular shapes could be broken up into triangles as well, which made the use of a triangle very obvious to me at least. In addition, the only real geometric math I use in my life (granted, I'm not an engineer), is based on triangles. – Sidney Jan 5 '15 at 16:50

There is a school of thought that holds that triangles are so big in the high school curriculum at least partly for historical reasons.

Firstly, triangles are big in Euclid.

Secondly, trigonometry was important to the military of the 19th century imperial powers in mapping and navigation.

Now, the circular functions are very important within mathematical analysis - you'll see a lot of them when you do calculus later in high school.

If we think of applications, geometry can also be approached through use of coordinate geometry.

Which approach is more practical depends on the starting information. If you are a surveyor using a theodolite, trigonometry is the technique to use.

But, if you are writing software or using GPS, your starting point is coordinates not angles. In that case, coordinate geometry using scalar and wedge products has many benefits. However, this is a new situation; it takes a long time for the curriculum to adapt.

I'd also note that the Pythagorean Theorem is not only famous, but perhaps the first theorem you see proved. Theorem proving is what mathematics is actually about.

• Coordinates (latitude/longitude) are angles... – Nick Matteo Jan 5 '15 at 3:12
• @kundor Not all coordinates are angles, but you are quite right in that lat/longs are. Although one does need cos to get locally flat coordinates from these, my point is that there is an alternative to traditional trigonometry for analyzing say plots of land. – Keith Jan 5 '15 at 5:21

Nearly everything in Euclidean geometry comes down to a divide-and-conquer approach:

1. Reduce the question to a question about triangles.
2. Use our extensive knowledge of triangles to answer it.

Several people have mentioned coordinate geometry, but ultimately that, too, comes down to our basic ideas about triangles. Without establishing those we haven't distinguished Euclidean geometry from, say, hyperbolic geometry, so we wouldn't be able to conclude anything nontrivial about our coordinates.

If there's a problem here it's not the material, but perhaps that students see this as a grab-bag of disconnected facts instead of the basis of all the math they're going to use in the future. Every single math class from here on out relies on a strong understanding of triangles, and that becomes more and more pronounced as things progress:

In calc 1 they'll learn to show that $$\lim_{x \to 0} \dfrac{\sin(x)}{x} = 1$$ using a geometric argument and the squeeze theorem, in order to determine that $$\dfrac{d}{dx} \sin(x) = \cos(x).$$

In calc 2 they'll need a solid grasp of trig so that they can take integrals like $$\int \dfrac{1}{1 + x^2} \, dx = \tan^{-1}(x).$$

In vector calculus they'll need to be able to convert between cartesian, polar, and spherical coordinates and understand why, say, $$dx \, dy \, dz = r^2 \sin \phi \, dr \, d\phi \, d\theta$$ and moreover they'll need to be able to parametrize a given region in three-dimensional space in, say, spherical coordinates.

In differential equations (or possibly a physics or engineering class) they'll need to grasp the Fourier transform.

In linear algebra they'll learn about inner products in finite dimensions.

Maybe it's not worth placing such a large emphasis on the topic. The Common Core standards about triangles are here, about 11 out of the 43 standards for geometry. I see at most one triangle center on that list.

Can you move on to other topics like similarity and areas sooner, rather than including them in the triangles unit? Then you can use triangles as a familiar example in the midst of a more interesting discussion.

• "CCSS.HSG.CO.C.10.(corestandards.org/Math/Content/HSG/CO/C/10) Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point." I would consider "the medians of a triangle meet at a point" a triangle center. – Xi Yu Jan 4 '15 at 18:46
• @XiYu, good point, I've edited to reflect it. But it's a pity: that and the trigonometry and the constructions are all topics in the standards that I'd like to remove. – user173 Jan 4 '15 at 19:10
• Thank you! (Now I'm just nit-picking, but incenter and circumcenter are also mentioned!) – Xi Yu Jan 5 '15 at 0:04

The reason triangles are so prevalent in geometry is because we work with a Cartesian coordinate system (XY planes/graphs). It's how we measure distances, by breaking them into two numbers. Three numbers if we're working with 3 dimensions.

So in order to reach something from (0,0) to (a,b) we need to have an X direction and a Y direction. The total distance is found out by the Pythagorean theorem, a third number. What do three lengths make? A triangle!

Geometry is the math between these relationships: the x, the y, and the distance. Add in oscillations around a circle, and you have your sine and cosine functions representing the X and Y coordinates. Check out the Trigonometry wikipedia page and notice how many triangles there are. We have an X, a Y, and a distance.

My humble opinion;

It seems to me that three factors might stand out for the prominence of trigonometry in mathematics:

• A (natural?) tendency to represent our 3D spacial experience in 2D; Up until a few centuries ago, the majority of our real word representations (paintings and drawings) have been two dimensional. I think that's around 50000 years of 2D doodling.
• The low cost of 2D communication; We only need sand and a stick to discover/develop, record and transmit spacial ideas (I wonder how this relates to the previous point).
• The enormous power of trigonometry; Both in our 2D representations and 3D experience, the primordial concepts of location and distance seem to be perfectly captured (and perfectly translated between them) with trigonometry.

So I guess I see it as the cheapest and most effective way to communicate regarding the space we experience.

In regards to its teaching value, and taking into account my ignorance in education theory, the fact that all of you are tired of so much trigonometry convinces me that it is overdone. I would hope that the curriculum could be shortened (leaving out simple stuff that can be developed/researched when needed if the basics are known). Surely exposing other areas (such as non-euclidean geometries) could provide students a higher level insight that would compensate any reduction in trigonometry curricula?

Maybe it has to do with congruence proofs. Proving line segments congruent is too uncomplicated and proving quadrilaterals congruent is too complicated (at least until after one has studied proving triangles congruent). Note that proving circles congruent is also too uncomplicated.

Here is a quote from a paper in The Mathematical Intelligencer of a mathematician who is in love with triangles:

No object has ever served mathematics better or longer. Compare the number of nontrivial results which are true for all topological spaces, rings, groups, etc, without putting extra assumptions on them with the number of nontrivial results which are true in any triangle...When it comes to deducing results in mathematics just from the definition of an object, nothing can hold a candle to the triangle. The triangle will serve mathematics forever.

It explicitly tells the value of triangles, however, it does not say anything about the way that this value could be appreciated in schools.

PS. This answer is a part of my wiki answer to a MO post. Thus, I've made this one a wiki answer as well.

Geometry fan here! There's so much stuff about triangles to learn. For example we have at least 5000 triangle centers compiled by Clark Kimberling and still being updated. That said, there's so much stuff with triangles that doesn't work as well for every quadrilateral, hence why whatever section on quadrilaterals always divides it into parallelograms, trapezoids, rhombuses, rectangles, sqauares, kites, etc. There's so few that can be said about any general quadrilateral besides Newton Gauss line and Miquel point, but all these require the knowledge of circles and of power of a point, whose proof relies on similar triangles. If you wanted to study only cyclic quadrilaterals you need to study first circles (done in the final 1/4 of the year) which itself requires the knowledge of right triangles.

It's not just polygons that can be triangulated, but any writer/director of films. This is a lithograph hanging in my home office.

• Technically, I don't think the picture is triangulated. – pkr298 Jan 6 '15 at 16:05
• Additional lines added to provide a shading effect. Else, the head looks triangulated to me. – JTP - Apologise to Monica Jan 6 '15 at 16:26
• Right below the nose and above the upper lip, is a pentagon. – pkr298 Jan 6 '15 at 17:42
• @pkr298 - Hmm. I see that, and I appreciate your diligence. You are right, it's not 100%, and a mod can pull my answer if they feel it too off topic. It seemed interesting to share to this topic as it hangs on the wall over my monitor. – JTP - Apologise to Monica Jan 8 '15 at 2:16

I just happen to visit here. I am not a math-educator. It is my opinion that trigonometic functions - which comes from triangle - uses in the broad range area of applications. Even in the mathematics geometry and analysis requires them in order describe or prove something. For example, Fourier serirs and Fourier transforms are so important in the digital world which it is hard to survive without it today.

I like this article that helps explain why studying triangles is important. "But triangles aren't just mathematically significant, they are also fundamental to the way we build our environments, both physical and virtual." https://wild.maths.org/power-triangles From my experience teaching, Common Core is attempting to raise the ability of students' analytical and reasoning ability by requiring more explanations (sentences) along with the numbers that pop out at the end--What do the numbers actually mean in the context of the (story) problem?

We can certainly do a lot to enliven discussions about triangles (and more generally polygons) in both K-12 geometry classes and college geometry classes. Looking at different ways to measure distance in the "plane" consisting of ordered pairs of real numbers is one approach to doing this. For example, after discussing taxicab distance one talk about whether in the taxicab plane triangles can be equilateral even when they are not equiangular. In axiomatic terms the taxicab plane does not obey the congruence axiom usually used in the Euclidean approach so there are many aspects of triangles in the taxicab plane that students have to get used to.

Classically, in the US, high school geometry has had a heavy proof emphasis (basically from Euclid). And most of that is about triangles. [Except make sure you hit the vertical angle theory hard. SAT test LOVES that concept and it is useful in a lot of math contest problems.] In this manner it is really sort of different from algebra 1 and 2, trig, analytic geometry, precalc, calculus, where we are really learning a lot of tools applicable to solving real world word problems.

Yes, there is a practical mensuration aspect of geometry also (formulas for areas and volumes and the like). And those end up being important for AP calculus (darned related rate word problems with geometry tricks like in the Stand and Deliver cheating problem!) But really, it feels like we have this horse that wants to run down the calculus track and we hank it off into proof land. And then getting back into mensuration formulas is short shrifted because that is yanking it back onto the type of thinking and algebraic problems solving we yanked away from! Don't know the answer as there is good argument that the emphasis on proofs in plane geometry is a good way to get some proof solving and reading experience that is missing or more difficult in the other subjects.