# What is the preferred way to denote the Pythagorean theorem equation?

I am teaching 12-16 year olds.

How should I write down the Pythagorean theorem equation?

Some alternatives:

$$a^2 + b^2 = c^2$$

$$\text{leg}^2 + \text{leg}^2 = \text{hypotenuse}^2$$

$$\text{leg}_1^2 + \text{leg}_2^2 = \text{hypotenuse}^2$$

$$b_1^2 + b_2^2 = h^2$$

$$x^2 + y^2 = z^2$$

$$AB^2 + AC^2 = BC^2$$

$$\text{(longest side length)}^2 = \text{(shortest side length)}^2 + \text{(remaining side length)}^2$$

I know it is context dependent, but say I want to calculate the length of the diagonal in a rectangle, or say I just want to refer to the Pythagorean theorem in general.

Edit: Based on the comments I would also like to suggest

$$\text{(First leg)}^2 + \text{(Second leg)}^2 = \text{(Hypotenuse)}^2$$

2nd edit:

I assume that the students have already seen and worked with the statement of the Pythagorean theorem in some variation before.

• Many of them have already seen a^2 + b^2 = c^2, which I'd guess is a standard. The form I also often use (not listed) is x^2 + y^2 = r^2, and I love to ask why the equation for a circle would be the same as something used with right triangles. In geometry class, I have sometimes written your last version. What drew you to the others? – Sue VanHattum Apr 12 at 14:15
• I almost always used $(\text{leg})^2 + (\text{leg})^2 = (\text{hyp})^2$ as a short-hand template in class, at least when it was understood that the legs did not have to be the same length (which was never an issue). To me several of the other forms seem overly formal for their intended use. – Dave L Renfro Apr 12 at 14:29
• I often struggle with students using $a$ and $b$ as the known length and $c$ as the unknown length independent of whether the unknown length is the hypotenuse or a leg. So I can see the usefulness of $\textrm{(leg)^2+(leg)^2 = (hyp)^2}$. – Steven Gubkin Apr 12 at 14:42
• @DaveLRenfro I think that I am not comfortable with "as long as it was understood that the legs did not have the same length". My fear is that they won't know that $x^2+x^2 = z^2$ can be contracted to $2x^2 = z^2$ when doing algebra, or that they will write $2 (\text{leg})^2 = \text{hyp}^2$. – Improve Apr 12 at 15:01
• $\text{leg}^2 + \text{leg}^2 = \text{hypotenuse}^2$ is quite bad (same leg, as commented above). $\text{leg}_1^2 + \text{leg}_2^2 = \text{hypotenuse}^2$ is ok. Letters or segment names without a picture are meaningless. Which is why the generic way to tell it in my country uses textual form: "sum of the squares on the lengths of the catheti of a right triangle equals the square of the length of the hypotenuse." – Rusty Core Apr 12 at 17:39

# Common knowledge

The formula $$a^2+b^2 = c^2$$ is common knowledge and the words for hypotenuse and leg (is "cathetus" not used in English?) are basic mathematical vocabulary. Including these seems a good idea.

# Connections to other mathematics

The notation with AB, CA and BC might be something the students have used or will use in less analytical geometry. Maybe you have an opportunity to mention that other context or tie these together, now or in a geometrical context.

# Jargon

Using some formulation without too much jargon is recommended; all the variables might be as much nonsense for some pupils, so this speaks for including some formulation that uses more natural language. This would also give you the opportunity to discuss why we use letters as variables in place of words (note that this is commonly not done in programming, for example; mathematics is peculiar here and an explanation might be order).

This also suggests avoiding needlessly difficult notation like subscripts, unless you feel that the students could use practice there and are ready for it, and would not have too much difficulty with Pythagoras. All extra cognitive load makes learning the main subject harder.

# Conflicts in notation

As mentioned Chris in his answer, $$h$$ already has a different meaning in the same context, so you might want to avoid this. Think backwards and forwards to see if there are other unfortunate notational conflicts.

# Curriculum

What does the curriculum that binds you say, if anything? Or maybe some other relevant guidelines. They probably don't go into this much detail.

# What does research say?

You might want to a quick literature search on for example Semantic scholar (or other academic search engine of your choice), search for something like "pythagorean theorem didactic", pick the first few articles that are accessible and seem relevant, skim through them and check if they discuss the matter of notation or link to something that discusses it. It is very possible to not find relevant stuff right away, but worth a few minutes, at least.

You can spend as much or as little time as you want on this, but at least skimming an article or two might also give other ideas on your teaching, so why not?

# In the end...

You have to check how much time you have and what the most important concerns for your students in the situation you are in. You can't do everything.

• I've never heard the word "cathetus" used in English, though I'd be unsurprised if it used to see use and went out along with "ordinate" and "abscissa", now universally known as $x$-coordinate and $y$-coordinate. It's interesting that "hypotenuse" somehow survived this de-Greeking process! – Kevin Arlin Apr 12 at 23:43
• @KevinArlin What alternative non-Greek name would you use then? "Arm"? – Federico Poloni Apr 13 at 6:33
• I have never heard the term leg used. I am a UK maths graduate. A triangle has sides. I have seen all the others but the picture would be forst – mmmmmm Apr 13 at 17:07
• @mmmmmm Apparently the British for "leg" is simply "one of the shorter sides" or "one of the other two sides." Kinda interesting. – Tanner Swett Apr 14 at 12:00
• Maybe a new question about this terminology issue might be interesting. It is wasted here in comments. – Tommi Apr 15 at 8:00

In Olympiad geometry, $$a$$, $$b$$, $$c$$ is the so-called standard notation for the sides of a triangle, so it makes sense to use it consistently when referring to a triangle (in isolation). However, in any case the general principle is introducing all your notation. Writing

Pythagoras' theorem states that $$a^2+b^2=c^2$$.

or

Pythagoras' theorem states that $$leg^2+leg^2=hyp^2$$.

is sloppy and should be avoided. Try to force yourself to take the high road and write

For a right triangle with legs $$a,b$$ and hypothenuse $$c$$, Pythagoras' theorem states that $$a^2+b^2=c^2$$.

If you are using the formula in a longer problem involving a rectangle, usually there is already notation that has been introduced earlier:

Let a rectangle be given with length $$l$$ and width $$w$$. Let $$d$$ be its diagonal. [...] By Pythagoras' theorem, $$l^2+w^2 = d^2$$.

• How has this worked for you with 12-16 year olds? I fear it might be a bit wordy; maybe the variables in a picture or in some other representation could also work. – Tommi Apr 13 at 10:47
• Adding a picture is a good idea for that age group; but this only reinforces "make sure your notation is understood" – Guntram Blohm Apr 13 at 11:07
• @Tommi I definitely did not try with 12-yo, but only with the top of that age group (15-18 range, Italian high-school). I agree with you that pictures may be more appropriate; my main point is to try to have variables named and introduced explicitly. – Federico Poloni Apr 13 at 11:11

Two alternatives I have seen used (am not necessarily recommending them, but will list some pros and cons) which don't seem to have been mentioned yet.

## Don't denote it algebraically at all! Draw a picture instead

For the lower end of the 12-16 age range, I've seen this work really well. You literally draw the squares sticking out from the triangle. Write the areas of the squares inside the squares. The two smaller squares add together to make the big square. • Even students with weak algebra skills can excel at this topic, at least at the three most common questions for beginners (use two legs to find hypotenuse; use leg and hypotenuse to find remaining leg; use three sides to verify if triangle is right-angled).
• Kids seem to enjoy drawing it out! It makes the topic much less intimidating and emphasises the shape/geometry aspect rather than the potentially off-putting algebra.
• Two of the most common mistakes with Pythagoras are (a) mixing up +/- depending on whether we should be finding hypotenuse/remaining leg; (b) forgetting to square-root at the end. Even among very weak students, drawing a picture seems to almost completely eliminate error (a) - it's obvious which the two smaller squares are, which should add up to make the big one - and substantially reduces error (b).
• Conceptually, makes clear the symmetry of the Pythagorean theorem with respect to the triangle's legs, with neither such finickity notation as $$\text{leg}_1^2 + \text{leg}_2^2$$, nor the potential to mislead of $$\text{leg}^2 + \text{leg}^2$$.
• Reinforces to students the geometric meaning of squaring and square-rooting. This is pretty useful among students around 12 years old, who by this stage are usually conceptually familiar with addition, subtraction, multiplication and division, and who generally see squaring as a special case of multiplication, but who often struggle with "what does square-rooting mean?" Those who try to see it as a special case of division often make mistakes like treating square-rooting as equivalent to dividing by two. Doing lots of drawing of squares while working through exercise questions on Pythagoras is an excellent want to build familiarity with what's at this age a rather alien operation.

Drawbacks

• Miss, or rather postpone, a chance to practise algebra and show how different areas of maths (algebra/geometry) are interlinked.
• Not so good for more abstract rather than numerical questions, e.g. when the side lengths are given only as algebraic expressions or you're applying Pythagoras as part of a mathematical proof e.g. of the law of the cosines.

Clearly if you want students to deal with more advanced material later, you can't stick to non-algebraic approaches forever. But I think "just draw a picture" is at least worth considering as an introductory teaching strategy for younger or less confident groups, for the first year that they see it. A potential time to switch to an algebraic notation might be when trigonometric ratios are introduced, since by this stage you probably want to be applying algebra to triangle problems. I say "probably" rather than "certainly" because there are, in fact, advocates of using non-algebraic approaches even to sin, cos, tan, e.g. by using "formula triangles". Again, this has pros and cons. But if you are going to label the sides of a right-angled triangle as the adjacent, opposite and hypotenuse, this gives you an alternative way to write the Pythagorean theorem.

## Denote the two legs as the adjacent and opposite

For example, if you use three-letter abbreviations for the sides, you can put:

$$\text{opp}^2 + \text{adj}^2 = \text{hyp}^2$$

• Consistent with how you teach trigonometry.
• Makes it unambiguously clear that the hypotenuse is the side that goes on its own side, which e.g. $$a^2 + b^2 = c^2$$ doesn't.
• A good starting point for teaching trigonometric identities. For example, dividing by $$\text{hyp}^2$$ produces easily recognised fractions like $$\frac{\text{opp}^2}{\text{hyp}^2}$$ and so we get $$\sin^2 \theta + \cos^2 \theta = 1$$. You could use a notation like $$\frac {a^2}{c^2}$$ instead, but it doesn't tell the story quite so clearly.

Drawbacks

• In principle, the Pythagorean theorem doesn't distinguish between the two legs, and it shouldn't be necessary to label an angle of interest so that the opposite and adjacent can be distinguished. In practice, once trig is taught, many schools drill students to mark on their angle of interest and label the three sides as opp, adj, hyp as soon as they see a right-angled triangle, so this may not be a serious flaw. (Arguably it's harmful conceptually to hide the symmetry with respect to the legs, but I doubt students will be hampered by this in their final exams. If they learned Pythagoras by the "just draw it" method beforehand, they may already be well aware of this symmetry anyway.)
• Depending on the structure of your curriculum, trigonometric ratios might be taught several years after the Pythagorean theorem, so it's a long wait until this notation becomes available to you.
• My only concern with this is that because of complexity of the diagram, the relative important of the right-angle marking seems diminished, and readers may easily start trying to apply it to triangles without two perpendicular sides. – Ben Voigt Apr 13 at 23:12
• So I post an answer that says "Don't denote it algebraically; Draw a picture instead" and it gets three downvotes, and later someone posts this and gets three upvotes. – Michael Hardy Apr 13 at 23:19
• @MichaelHardy This answer discusses pros and cons and shows knowledge of the age group the answer is targeted for. – Tommi Apr 14 at 6:04
• @Ben Applying Pythagoras (and sin/cos/tan ratios) to non-right-angled triangles is a common mistake regardless of method! Fwiw, I haven't noticed it being more common in students using a drawing method. At least (re-)drawing a diagram, usually in a standard orientation (legs horizontal/vertical with right-angle at the base) gives some visual importance to the right angle. Students who jump into plugging numbers into a formula without drawing a diagram first are still prone to pick an inappropriate formula - lots of teachers suggest students draw a diagram before using formulas for that reason – Silverfish Apr 14 at 12:01
• @Ben I do think "right-angled triangles only" is an important point to emphasise though. One way to drill it home is to set exercises asking students to classify triangles as acute/right-angled/obtuse depending on whether the sum of the smaller squares is more/equal/less than the larger square, even if that's not explicitly covered in your syllabus. That should reinforce why we can only use the equality if there is a right angle - but of course, doesn't mean students won't sometimes apply it inappropriately regardless! – Silverfish Apr 14 at 12:05

The only one of these that looks objectionable to me is the one that calls the hypotenuse $$h$$, since in a triangle the letter $$h$$ usually refers to the triangle's height (which could be either one of the legs but could not be the hypotenuse).

• A similar issue arises when calculating the volume of a pyramid with a triangular base. Then $h$ is sometimes used to denote both a height in the triangle and the height of the pyramid. – Improve Apr 12 at 15:04
• Why not be precise and use base versus altitude? One side versus the triangle-crossing-distance at right angles to it. Using height for a triangle directly confuses students when none of the relevant measurements are vertical. On the other hand, if they're advanced enough to handle height meaning also a non-vertical measured dimension, they're more than capable of accepting that one symbol may be defined with multiple meanings in separate contexts. – Nij Apr 13 at 10:34

Whatever you choose, make sure to follow some basic rules:

• Clearly state the preconditions and make sure they are understood (right triangle in your case).
• Clearly state the meaning of the symbols (e.g. which symbols stand for the sides adjacent to the right angle, and which for the third one).
• Use the symbols consistently (don't make e.g. the same symbol "leg" stand for two different sides with two different lengthes).

Being sloppy at these topics creates lots of problems later on (students just guessing when to apply some formula, students not understanding the notion of a variable, ...).

I really wish people would stop teaching the Pythagorean Theorem as $$a^2 + b^2 = c^2$$, for the following reason: Give your students the diagram below, and ask them to solve for $$c$$. At least 1/3 of a typical high school class will write $$a^2 + b^2 = c^2$$ and report back to you that $$c = 5$$. The problem is that the equation $$a^2 + b^2 = c^2$$ is so memorable, so ingrained in them, that automaticity kicks in and overrides any consideration about what the letters actually are supposed to refer to. I much prefer a semantic description such as $$(\textrm{first leg})^2 + (\textrm{second leg})^2 = (\textrm{hypotenuse})^2$$ • I don't find this reason convincing. I think the fact that the formula is easy to remember trumps the fact that students can be fooled by labeling decisions. Also, I don't feel like this sort of tricky labeling will show up very often in real-world applications. – Brian Apr 15 at 17:39
• +1 In software development we have long overcome the habit of using single-letter names, and IMHO math would benefit from more descriptive names as well. – Ralf Kleberhoff Apr 16 at 7:00
• A^2+B^2=C^2 isn't actually a definition of Pythagorean theorem (or at least, not a self-contained definition). It's just an especially effective mnemonic that people pretend is a definition. So, I think it would be better to compare it to other mnemonics (e.g., sohcahtoa), rather than to proper definitions. Mind you, none of this is a reason not to treat it as a definition (i.e., by defining a, b, and c) for purposes of teaching. – Brian Apr 16 at 21:45
• @brian I disagree that it is just a mnemonic like sohcatoa. The names of the three primary trigonometric ratio are just that -- names. There is no logic to why opposite/hypotenuse is called "sine"; if you forget which ratio is which, there is no way to figure them out. So mnemonics in that case are necessary. In the case of $a^2 + b^2 = c^2$, my experience is that for many students the easily-remembered mnemonic takes the place of, rather than reinforces, the actual statement of the theorem. It does more harm than good. – mweiss Apr 16 at 22:09
• I think this is a perfect reason that you should state it as $a^2+b^2=c^2$ (and of course you draw an appropriate diagram each time). Then you give them the problem you showed and if 1/3 answer incorrectly, that's a perfect learning opportunity. To answer $c=5$ demonstrates a tenuous grasp of mathematics that should be dealt with head on. Trying to state theorems to avoid exposing existing gaps just sweeps the problem under the rug. – Thierry Apr 17 at 1:36

I prefer to state it the traditional way, not involving multiplying a length by itself, but involving areas of those polygons known as squares.

Draw two lines at right angles to the hypotenuse at its two endpoints to get two sides of the square other than the hypotenuse itself, and draw the fourth side, and there you have a square, whose area is the sum of the areas of two squares having the two legs as one side.

• Would you do this whenever you applied the Pythagorean theorem or referenced it? – Improve Apr 12 at 21:57
• @Improve : No. I do it when I explain what the theorem says and when I present a proof of it. – Michael Hardy Apr 12 at 22:01
• (+1) I didn't notice this answer at first, due to the downvotes. My answer includes a variant of this (writing the areas inside the squares) I've seen successfully used in high school settings. As a frame challenge to the original question, I think this answer has several advantages: algebra is not the only way to represent geometric facts, and not always the best one pedagogically. I suspect 12-year-olds encountering Pythagoras for the first time are likely to grasp the theorem and its applications far faster from a picture than from being presented with $leg_1^2+leg_2^2=hyp^2$ – Silverfish Apr 13 at 14:28
• @Improve : I wonder whether you are among those who downvoted this. This reaction seems quite mysterious to me, since it says the same thing that is at the beginning of "Silverfish"'s answer. – Michael Hardy Apr 13 at 23:30
• @MichaelHardy I neither upvoted nor downvoted this answer. If I were to introduce the Pythagorean theorem to my students, writing down the standard equation would be step 8 and not step 1, so I agree with your sentiment. However, I am not convinced that the standard equation has no role when solving problems, and I was hoping to find a good way of writing down the equation. – Improve Apr 13 at 23:50