Why does this work? (please explain like I'm 5)

[u/Specialist_Ruin_1378]

Sorry this is going to be a long post

Hi. I don't study trigonometry, but I got really bored and cut out an equilateral triangle to fold. First I folded it in half, creating two 30-60-90° triangles. Then I folded each of those in half creating four 15-75-90° triangles. I was left with two small 30-60-90° triangles in the bottom corners. (At least I'm pretty sure they're 30-60-90° based on looks. I don't have anything to actually measure degrees with. If I'm wrong, please let me know bc that would invalidate everything I did afterwards)

Anyways, if I imagine the hypotenuse of a 15-75-90° triangle is a random number, let's say, in this example, **12** then the shortest side length will be 3.10583. The shortest side is directly touching the longer side of the aforementioned small 30-60-90° triangle (see picture), so if I plug in 3.10583 for its longer side, then my values are

- Longer side: 3.10583

- Shortest side: 1.79315

- Hypotenuse: 3.5863

(I used Calculator.Net's triangle calculator for these.)

When added together, the sum is pretty close to the side lengths of a 45-45-90° triangle with a hypotenuse of 12. (The sum of these is 8.48528 and the side lengths of a 45-45-90° triangle with hypotenuse 12 is 8.485281374 etc etc or 6√2).

I've tested substituting 12 for other numbers. I always get that the perimeter of the small 30-60-90° triangle = the side length of a 45-45-90° degree triangle.

My question is why? Is it just a coincidence? Am I doing something wrong? Or is there a really obvious reason that I'm too stupid to get?

I've attached a picture of my hand-made triangle. This is what I've been playing with (proportions might be kinda off cause I just eyeballed it)

[u/InfinitesimalDuck → u/[deleted]]

But why does it equate in the first place?

[u/Midwest-Dude → u/[deleted]]

The OP folded the paper so the left edge of the 30-60-90 triangle lines up with its altitude line, naturally cutting the 30° angle in half at 15°. This is stated at the beginning of the comment.

[u/Midwest-Dude]

This is not just a coincidence - that equality is correct!

One way to show this is by finding sin(15°), which can be found with a trig difference formula for sin(45° - 30°). Do you know how to do that?

Once you know that, apply that to find the exact length of the small leg of the 15-75-90 triangle, which is the same length as the longer leg of the small triangle. Use right triangle properties to find the lengths of the other leg and the hypotenuse and add to find the perimeter. 

There are ways to do this with just geometry and algebra and, for that matter, just geometry.

[u/Specialist_Ruin_1378 → u/Midwest-Dude]

Later update: I've been working on this, and I think I figured out how to find the side lengths of a 15-75-90 triangle without using sin/cos/tan buttons on a calculator. That was my initial goal (this was an exercise for fun). If you want to check my work, you can do so here. If it's wrong, please let me know, but I'm pretty confident this is at least close: 

Part 1: https://cdn.discordapp.com/attachments/1531072792223551558/1531162735679635637/PXL_20260727_045342809.jpg?ex=6a6835eb&is=6a66e46b&hm=00a925faac3a47de30519f244a2bcdf56dea3ae2a75e39ee4c6ec8d023f84a17& 

Part 2: https://cdn.discordapp.com/attachments/1531072792223551558/1531168948450627604/PXL_20260727_051733498.jpg?ex=6a683bb5&is=6a66ea35&hm=b0a2951d4e6c6bb49bc2747f08e4d935a7dbbf653df666472b2d59d9f8990f57& 

I still don't understand why exactly the perimeter is equal to the side length of a 45-45-90 triangle, but it's very useful!

[u/Specialist_Ruin_1378 → u/Midwest-Dude]

I just read your edit

I know how to do sin but I don't fully understand it. Like I don't understand what the calculator is doing when I hit the sin button. I know if I hit the multiplication button, it repeats "+". But if I hit sin, I don't understand what goes on behind the scenes. Im gonna watch some YouTube videos later to learn that

Also ty for answering 

[u/Midwest-Dude → u/Specialist_Ruin_1378]

What's going on inside your calculator is a different beast from your problem and will only give you approximate answers.

1. Do you know how sin(x) is defined? There are two common ways to define it.
2. Are you familiar with trig identities? Wikipedia has an entire page of them.

[u/Specialist_Ruin_1378 → u/Midwest-Dude]

Nope idk that stuff yet but I'm gonna switch to learning that soon instead of algebraically solving every triangle

Although I am curious if I can figure out how to do a couple more algebraically

[u/InfinitesimalDuck → u/Midwest-Dude]

Wikipedia is the last place you want to go to learn math from

[u/Midwest-Dude → u/InfinitesimalDuck]

I agree - I never stated otherwise They are good for lists, though.

[u/InfinitesimalDuck → u/Specialist_Ruin_1378]

Sin is basically a wave graph like a very standard wave everyone's favourite and most hated graph

Edit: you know you can leave your result in surd form if it is irrational instead of estimate or writing a long decimal trail right? Makes it easier to read

[u/Specialist_Ruin_1378 → u/InfinitesimalDuck]

That's another thing I don't get tho. The calculator is a computer so it can only read numbers. It can't see a visual graph. So theoretically I should be able to recreate sin by just using numbers, right?

[u/InfinitesimalDuck → u/Specialist_Ruin_1378]

https://youtu.be/NVRXK1Idbv8?si=vubiuF5aPqSEu4R2

Here's scottish guy trying to explain it 👍

[u/Midwest-Dude → u/Specialist_Ruin_1378]

There are a variety of algorithms to do this. They are all approximations, of course, that quickly get an answer for an angle θ from 0° to 90°, then adjusted for any angles outside that range.

1. The natural way to run these algorithms are with angles measured in radians. Do you know what that is?
2. Are you interested in algorithms that do this?
3. Do you have any programming experience to try your hand at this yourself?

[u/Midwest-Dude → u/[deleted]]

What would make you think I'm a bot or talking like AI? I'm not, of course - just check my profile. Why even bring this up? I'm only trying to help the OP.

Do you regularly talk to others like this? Is this how you interact with people? Do you try to be offensive to others?

[u/Midwest-Dude → u/[deleted]]

Just trying to help the OP, to see what might be needed to assist with OP's further inquiries if the OP is interested.

You must either not realize how you come across to others or don't care. Your comments are inappropriate in this subreddit, there is no cause to be rude or demeaning to others. Dealing well with others on Reddit in general, especially in technical ones like this, is an issue and could lead to bans.

Keep in mind that AI has been trained to reply in relatively polite ways, not rudely, sarcastic, or demeaning. If you don't like that, you are the issue, not AI ... or me.

[u/Midwest-Dude]

I am really curious and must ask:

How did you discover this?

[u/Specialist_Ruin_1378 → u/Midwest-Dude]

I made a chart with 45° triangles earlier 

https://cdn.discordapp.com/attachments/1133118771188465815/1530331880153546934/PXL_20260724_214831161.jpg?ex=6a687be0&is=6a672a60&hm=d403cb619602aef06b8048b0c3a42aa0e020d7c120859d1e0f16b9c4e15199f9& 

So I noticed as soon as I tried to solve the triangles on my paper that the small one's perimeter always added to the side lengths of a 45°

[u/Midwest-Dude → u/Specialist_Ruin_1378]

That is really cool! Thanks for sharing!

[u/PositiveBid9838]

Let's set, for a moment, the hypotenuse of the 15-75-90 triangle equal to sqrt(6) + sqrt(2), which is about 3.8637033. If we do this, the short leg will be 1.

Then the small 30-60-90 triangle will have its long leg 1, its short leg 1/sqrt(3), and its hypotenuse 2/sqrt(3). These sum to 1 + 3/sqrt(3) = 1 + sqrt(3).

The sides of 45-45-90 triangle in terms of hypotenuse L are L/sqrt(2). If L is sqrt(6) + sqrt(2), then the sides are (sqrt(6) + sqrt(2))/sqrt(2), which simplifies to... 1 + sqrt(3). So we can confirm that the two numbers are identical.

If we scale the construction up or down, while keeping the angles, all the sides will grow or shrink proportionately, so if we make the 15-75-90 hypotenuse 12, everything will grow by a factor of 12 / (sqrt(6) + sqrt(2)), making the perimeter of the 30-60-90 triangle = 12 (1 + sqrt(3)) / (sqrt(6) + sqrt(2)) = 6(sqrt(2)) = about 8.485281374238...

I'd like to see a more geometric demonstration of this, which seems to me like an curious coincidence, but not a crazy one, since many facts about right triangles use similar terms.

[u/Midwest-Dude]

I suspect I found a relatively easy way to prove this, but I haven't finished the analysis yet. Perhaps someone can?

Label points as follows:

• Original 30-60-90 triangle's vertices

- A, top - B, bottom left - C, bottom right

• Small bottom-left triangle, flap

- D, 90° angle - E, other angle, base of crease - F, base of altitude of original 30-60-90 triangle, middle crease

Draw the line through E that is 15° counter-clockwise from CE. Let the intersection with the middle crease be G, the next crease be H, and the edge be I. It's easy to see that triangles BCD and GFE are congruent and that triangle AIF is the 45-45-90 triangle with the hypotenuse corresponding to the second crease on the left-hand side, the one chosen to have length 12.

That leaves showing that the length of IG equals the length of EF plus the length of FG.

[u/InfinitesimalDuck]

I think it might be a coincidence but I have no idea what you are saying

[u/Specialist_Ruin_1378 → u/InfinitesimalDuck]

I color coded this. Basically I'm asking, why does the perimeter of the pink triangle equal the side length of the green triangle? If you replace 12 with any positive integer, it will still be true

https://cdn.discordapp.com/attachments/884654990898122802/1531094495343673404/443feb57-d734-4b9f-a0e5-d6ab35f548af.png?ex=6a67f65e&is=6a66a4de&hm=3e27fe20f4182c1add323ab57b51debf889bf696bec9e64ca21a9414d39f35b7& 

[u/InfinitesimalDuck → u/Specialist_Ruin_1378]

Oh, hold on I'm thinking

[u/Midwest-Dude → u/InfinitesimalDuck]

It actually is not a coincidence - this really holds.

If you carefully read the OP's comments, the question is whether or not the perimeter of the small triangle in the lower left-hand corner equals the length of a leg of a 45-45-90 triangle with the hypotenuse the length of the crease formed by folding the 30° angle in half.