# Applications of High School Geometry

Sometimes I struggle to give my students a sufficient number of reasons why they should study Geometry in high school, other than that it helps them think and increases their understanding of the world. Aside from sports and teaching itself, what are examples of jobs/careers that apply this material?

• I wouldn't focus on careers. I'd talk about how it helps you learn to visualize. And I'd focus on making the course interesting. Feb 18 at 2:12
• May be ponder questions such as: Why do cars need differentials? What makes a lathe work? Ok, that was circles. May be include scissor lifts and such to get parallelograms and triangles into play? My god child enjoyed it immensely when I constructed extendable "scissors/pliers" for him from his legos. Not really career related, but... Feb 18 at 18:46
• Frame challenge: "Helping the students think" and "increasing their understanding of the world" are already two extremely good reasons to teach something in high school. Why should high school students read Shakespeare or Poe or Dante or Goethe other than that it helps them think and increases their understanding of the world? Feb 18 at 20:26
• Don't go down the slippery slope of made up reasons and motivations. It's all about learning to think abstractly. Prefrontal cortex development happens at about this age, and that's when abstract reasoning blossoms or dies. The slippery slope can end up as "why do I have to learn how to read if my computer can read text out loud to me." Of course I am being whimsical, but... Feb 18 at 23:37
• @JochenGlueck I am still not convinced about Shakespeare. I still think that was a waste of time. Feb 18 at 23:53

Every time I'm doing carpentry/DIY remodeling on my house, I use so much high school geometry that it makes me chuckle.

I've had instances where I need to cut angles to make pieces parallel and used Alternate Interior reasoning to pull it off. I've had times where the angle I can measure is complimentary to the angle I need to cut, so I have to figure out how to reflect the piece or the cut line to get it right. I've needed pieces of plywood cut that were general quadrilaterals (because nothing is square in old homes) and was able to pull it off by measuring diagonals and marking equidistant points using a string compass. And of course there's all the tricks for actually getting a square corner.

Of all the subjects taught in school from algebra to calculus to abstract algebra, I've used high school geometry far above and beyond the others in these types of "real" world situations. Set up an equation and solve for $$x$$? Not sure I've ever really had the need to do that outside of my professional life (okay, maybe a few times when I was baking). Need to literally construct a congruent triangle? I can't even begin to count how many times.

• I agree completely. I have an antique set of compasses that belonged to my grandfather, who was a carpenter. It is similar, if not identical to this. I have used it for some little household projects where precision was important. Feb 18 at 14:56
• A few years ago, I had my conservatory roof replaced. The fitter measured the heights at the front and back, and the dimensions of the floor. Then he went to measure the length of the roof, and I told him what it would be (with a quick bit of Pythagoras). The look on his face, you'd think I'd just pulled a white rabbit out of my mouth. It's kind of shocking how few tradies really know basic geometry which would be so valuable to their work. Feb 19 at 1:03
• There was a question a while ago - not sure if it was in math or garden&landscaping - from a landscaper: "My client wants a circular lawn, but the center is within the house. How can I achieve this?". Feb 19 at 12:06

Here's my rant that I gave to my students:

You are learning this right now because some of you will need this later in life. You may need this information for your daily job. You may need this information to check off a requirements to GET your job.

We don't know who will actually need to use this yet.
We do know that it will be MUCH harder to start learning this when you are all older.
We teach it to all of you now, while you are young and have an easier time learning.

Math education is a tower. You have to build one floor before you can start on the next. The floor of Geometry opens the door to higher math.

There are many jobs where you WILL need to reach for the top of this tower of mathematics. Many of them are good jobs.

We all benefit from the things those people create with their knowledge and experience.

Your phone could not exist without very intelligent, hard working people who most likely learned this math when they were young.

The internet, streaming video, video games... all require a ton of math to work.

Some of you may never need to worry about internal angles or parallel lines again after this class, but the world is better when everyone learns this now.

Even if you don't think you'll ever go into a career that requires it, you do not know what your future holds for you either.

If you do end up needing this, it really, REALLY sucks having to go back and start learning it in college.

• The problem is that the students come with very different learning styles and attitudes. Some learn what ever the "authority" tells them to learn, some find maths genuinely interesting, some don't and cannot motivate themselves to learn something without apparent need. The typical "bad at school, good at their job" people usually fall into the last category. Feb 21 at 10:59

The problem here is usually less a matter of ‘I know of no instances where I will use this’ and more ‘I cannot relate things I actually do to the abstract concepts here’. A very large number of things people do regularly actually use a huge amount of math, but they never think about that because they have trouble translating from the abstract aspects to concrete applications.

The best example of this that I can think of for geometry is actually shooter games. You can’t do accurate target-lead calculations without using trigonometry, and thus you need at least a basic understanding (be it formally taught or simply learned intuition) of trig to be able to reliably hit moving targets. Most people who play shooters regularly have such an understanding, but in many cases it’s an intuitive skill and they can’t explain it easily in formal mathematical terms, if at all. But, on the other hand, having a knowledge of trig can greatly accelerate learning that as an intuitive skill.

• Then you get to Minecraft speedrunning, where top players used to have to do geometry/trig calculations in seconds. Now they have programs to do some of it for you, but the person writing the programs is working with geometry, trigonometry, and statistics in order to get accurate results. Feb 19 at 17:46

[This may seem indirect, but I want to give you something different, therefore additive.]

Nate: For geometry, specifically, I think you have to tell the kids something along the lines of

"This course is going to feel different from algebra class...and it will feel different from the classes after, where you get back to algebra. In this class, we are going to do a lot of proofs. It's important that you see what it means to prove something logically. Even if you don't remember every picky part of geometry class, you will know what it means to prove something and know what that feels like. Geometry was a wonderful invention by the Greek philosophers and something that many important thinkers like Abraham Lincoln loved. Because they loved to think logically.

Some of you are going to love how this class is different from last year's algebra. And some of you are going to hate it...and want to get back to equations with letters and numbers. But...I'm just warning you so it won't be such a shock, this year is going to be very different from algebra."

Note: I'm actually way less proof-y than the average "want to teach real analysis in calculus class" MSE/MESE type. So this is a sinner counseling morality. But geometry is an exposure to proof. It just is. And even if it's wrong...that's what it is. And I don't really even think it's wrong or if it is, very wrong, to have an exposure to proofs.

Realistically, I think straining to find direct applications in daily work (or personal) life is not the right rationale for the course. It's a part of the overall math program, which is mostly (i.e. well over 50%) a tool for science and engineering, which there are a lot of jobs in.

For students that are college-bound in STEM, they need to learn a lot of math...and later math has some dependencies on geometry. Especially trigonometry, which is all over the place in ODEs, calculus, kinematics, electrical engineering, etc.

For kids that are college-bound in non-STEM, they will need less math overall, but still may need some (e.g. medical fields). And geometry will be a small part of that.

And all of them, college-bound or not, will need some basic grounding in math/science to be citizens and read the newspaper, with some amount of perspective. Yes, they won't all have perfect or deep mastery...and will forget a lot of it. Same could be said of history and geography...how many people have read Mahan and have some insights from him on modern geopolitics? All that said, if you don't know France is west of Germany is west of Poland...or that the Germans invaded Russia in WW2, you can't thoughtfully read NYT articles about NATO security.

P.s. I wouldn't transfer this explanation as is. But you need to be informed by it. Often I find these questions are more coming from instructors who are concerned themselves about the relevance of the courses, rather than because they really are too often challenged by students (95% of whom will take direction and intuitively know that the course content is fixed) and can't reply well. I would probably avoid getting too much into these relevancy debates. Especially don't open them if you don't know how to handle them! The kids have to do it because it is a part of school. Someone made a decision what is needed and that's that. You don't need to say it so brutally, but at least in your mind, know it is true. And don't let the one doubting Thomas derail your class.

Edit: (Hope I don't lose my, surprising, +1s for this) I'm not sure that you want to say this...or if you do, be careful how you package it. This is a little Machiavellian...but geometry is a pretty core part of the SAT-Math exam.

If we want to get meta, we can argue what's the point of the SAT (and there are good arguments pro/con). But if we just live in the rat race we live in...doing better is helpful to kids.

Now it's not really a pure subject matter test...has aspects of aptitude (in the title even). But it's not a pure aptitude test either. It certainly is a lot easier to take that test if you've had basic geometry (and basic algebra) than if you have to nuke everything out from pure smarts. I mean look at vertical angle theorem. ETS loves that...wants to make babies with that theorem.

• +1 Large parts of this never ending "practical usage of xyz"-debate are rather misguided because they miss many of the points in your answer. Feb 18 at 20:13
• +1 Also you mention college bound vs not. Early in high school, say grade 9 or 10 when geometry is taken, nobody knows which students are college bound! We do not sort students early on to "you will be a chess grandmaster, you will be a gymnast, you will be a mathematician,..." So we have to treat >all< students in geometry as college bound. Most students will not want to hear this, but it is good for the instructor to think about. Again..+1 on your answer! Feb 18 at 23:32
• +1 I am glad to hear that people still teach about proofs in their geometry classes. Feb 20 at 14:45
• If anything, it's too much. But it is what it is (not complaining). 80%+ is proofs. Death by congruent triangles. Maybe less than 10% (each) are construction (which seems a sop to tradition) and mensuration (area, volume...useful!) I also felt that the course sort spent too much time in plane geometry and ended up skimping on solid geometry. Also, sad that there was not even a "look at the animals in the zoo" type of exposure to some more interesting geometric objects (truncated pyramids and the like). I ended up getting that in mechanical drawing instead! Feb 20 at 14:56

Before you cut a tree down, it is quite useful to know where it will land, which requires calculating its height. There are several methods depending on similar triangles, e.g., Wikipedia article.

A variation was depicted in this classic Tom & Jerry cartoon:

• Although I have to admit I don't understand the woodpecker's calculation. Feb 18 at 16:49
• Take a stick the length of your arm or a bit longer. Hold it so that the length above your arm stands above your. Walk to where the top of the tree lines up with the top of the stick when your hand is at eye-level. You're standing about where the top of the tree will fall, if the tree falls in your direction. Assuming the ground is more or less level. OTOH, actual arborists take down trees in less risky ways than felling the whole thing. Especially when you have to worry about where it falls. Feb 18 at 17:02

In general, it makes it more difficult for someone to deceive you. It also lets you better recognize when you're getting a good deal.

It helps when you're buying something based on dimensions to determine whether the price is reasonable or not. The size might not be super critical so you might be willing to spend a bit more to go up a size to have some extra material if the cost per volume gets better for the larger size, or save money by going down a size because the cost per volume sky rockets for the larger size:

"Why are these 1/2" rods so expensive? They're nearly double the price of the 3/8"! What a rip off! Oh wait, 1/2" is only 33% more diameter than 1/2" but that means the cross sectional area is 76% more. I guess that's why it costs so much."

I've also seen people complain about that a 5" vise is a rip off because it costs double the 4" version of the vise. You can look at the photos of the two vises and notice that the 4" and 5" vises look identical. The 5" vise doesn't have any ungainly proportions because there is no difference in proportions. It might be obvious if I point that out, but how is a person uneducated in geometry going to notice this? That means the 5" vise isn't just like the 4" vise except only the jaws are 25% wider. It's that the 25% larger in all 3 dimensions which means it's 1.95x more material. THAT's why it costs double. Incidentally, that also means it's weighs double and you might not want to deal with that for only a 25% increase in jaw width.

It comes up when buying a house. It seems a lot of people only know of square footage by feel, or comparative numbers, or looking at other houses of known square footage. Especially for specific rooms, can just go to a kitchen, look down, notice that the tile size is 12" x 12" which makes it one square foot and just counting up a set number of those in particular sized rectangle just looking at that.

It also helps you calculate the best pizza size to buy to get the most pizza for your money.

It helps you figure out the shortest wood plank you can get away with while still being at an angle where the mouse can easily climb up it to get into your bucket mouse trap.

Oh, and perhaps the most important thing it does for you as far as your students are concerned:

For the girls, it helps prevent them from being easily impressed by rudimentary knowledge when, for their elective, guys from the second and third year of the engineering program start taking introductory STEM courses meant for first year art students as a way to meet girls. They would get closer to them by helping them with their homework and impressing them with basic knowledge. Your students may laugh, but this actually worked for multiple guys when it happened to the point there were complaints about other students about showboating. It's one thing to be impressed when someone solves the heat equation. It's another to be impressed when someone finds the area of a circle.

For the guys, it helps them meet and impress girls in the arts program who did not pick up this knowledge.

The answer to most "when am I gonna use this" questions is the same. The best analogy that I've heard is a gym. People go to the gym and do bicep curls with weights. Why? Are they ever going to need to lift something while they're somehow restricted to moving only the lower half of one arm? Of course not. They do it because that exercise isolates and strengthens a specific muscle group. By doing many such exercises, they develop a body that can use all of those strong muscles together to do all sorts of things that weren't part of their exercise routine. You don't go the gym and practice moving a couch up the stairs, you build up all the individual muscles so that when you need to move a couch up the stairs you can do it.

School is the same way. The classes you take and the activities you do isolate and strengthen specific parts of the mind. A strong mind can tackle tasks that are far beyond anything you could ever learn in school. To get there, though, you have to have a well-balanced exercise routine that increases in intensity as you grow.

Speaking specifically about high-school geometry, I've always ranked that class as the most important class out of my entire primary education. At one point, my teacher explained that the course really didn't have anything to do with shapes or angles or any of that stuff. The purpose of the course was to learn proofs and by extension, developing deductive reasoning skills. All the stuff about shapes and the other things we typically associate with geometry are all just a delivery mechanism for learning this core concept. They're easy to draw and visualize, which makes it easier to learn than if you were doing the same with a more abstract subject matter like algebra or philosophy. My teacher said that she didn't care if we left the class and instantly forgot how to calculate the volume of a cylinder. She was more concerned with whether we could approach a scenario by taking observations, applying logic and facts that we know to be true, and then arriving at a solution for a problem that we've never encountered before.

Looking at geometry from that lens, when does any of this apply to real life? Every time you read or hear something and want to figure out whether it's true. Every time something in your house breaks and you need to get it working again. Every time you need to navigate any sort of situation that's new or unusual. In other words, all day, every day. Many professions (medicine, law, engineering, software development, auto mechanics, etc.) are directly based on this type of deductive reasoning loop. Even outside specific job scenarios, it's a process that (hopefully) runs inside your head every time that you have to think your way through a problem. If you don't learn how to reason effectively, you'll be prone to making poor decisions and find it harder to continue to learn as you go through life. Or even worse, you might not have the courage to try and tackle new problems at all.

My wife is a quilter. We are constantly scaling shapes, figuring out how much fabric is needed to make something, finding the best way to cut fabric to make a given shape, computing how many squares to make a quilt of a desired size or the border to stretch a center to a desired size, and many more.

I think a big part of why I was successful with math in junior high and high school is because I had a strong interest in video game programming. It turns out that 3D rendering is a lot of just algebra, linear algebra, geometry, and trigonometry. There are many different career fields where this is relevant (video games, movies, architecture, etc), and the nice thing is that it can be a fun topic for kids to learn.

I do a lot of spatial computing now, and interestingly, many of the math concepts underpinning 3D rendering are applicable to GIS (Geographic Information Systems), which made learning spatial computing very easy for me. Specifically, one of the major types of data used in GIS is "vector" data, which consists of points, lines, and polygons. GIS is used in many professional fields (environmental sciences, infrastructure planning, economics, etc.). GIS expertise can lead to very well-paying jobs.

you can use history of mthematics for that.

Geometry has played a pivotal role throughout history in various fields, and understanding its applications can inspire students to appreciate its value beyond the classroom. Here are refined examples to illustrate this:

Ancient Land Measurement: In ancient Egypt, geometry was vital in calculating the area of land for taxation and re-establishing boundaries after the Nile's annual floods. This demonstrates geometry's role in governance and civil administration.

Astronomical Discoveries: The Greeks used geometric principles to estimate large distances, such as the gap between the Earth and the Sun. These methods laid the groundwork for modern astronomy and space exploration.

Cartography and Navigation: During the Age of Discovery, geometry was crucial for mapmaking and navigation. It helped explorers chart courses and map the Earth's surface, leading to the world as we know it today.

By presenting geometry as a historical tool that solved real-world problems and paved the way for current technologies, we can inspire students with tangible examples of how geometry shapes our understanding of the world. These historical applications not only show geometry’s practicality but also its enduring legacy in advancing human knowledge and society."

As is fashionable, I talked to my pupils about careers in artificial intelligence. I think they were interested. I will be told that logic that comes into play in AI is not geometry. I'll get to that. Today's mathematics doesn't pay much attention to classical geometry because we have so much to teach that we often don't have the time to do it. But getting students to study classical geometry is never a waste of time.

Everyone knows or should know that if classical geometry does not play a major role in modern mathematics, on the other hand the spirit of geometry reigns everywhere (the sentence is not mine, I don't remember who said it). It is perfectly illusory to learn anything in mathematics without some basic geometry. For example, if we consider one of the simplest structures, that of vector space, do not connect the elementary results from linear algebra to elementary geometry, it is to build on sand and condemn oneself to psitassism.

And how can we learn logic, in order to return to it, without having some geometrical results to illustrate its principles? (see here for example).

In short, there is no math without geometry. And since I've heard that $$75\%$$ of the jobs offered today more or less use math, I don't hesitate to repeat it to the students.