These are special triangles and they're called hero triangles.

You're like that, Hiromi In triangles on these are particular types of hero triangles.

Hero Triangle is a triangle that has introduced sites holder besides, on an interject area.

Okay, so it's just something that mathematicians like to play with.

You should think of these a lot like perfect numbers and things like that.

It's just something that mathematicians like that fun with.

We played with these for a long time.

Even now, we're still coming up with new things about them.

But these are particularly special.

The superhero trying These are superhero triangles.

Absolutely.

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Get Captain America over here Are Can't believe it's superhero triangles.

So my 1st 1 it's a right angle triangle.

So the idea is they've got into decides on So this has sides 5 12 and 13 which is called a Pythagorean triangle as well.

Pythagorean triples.

But this is why I wanted to point out Look at the perimeter of the perimeter is adding up the three sides.

And so I reckon the perimeter is 30.

So 12 plus 13 plus five on the area.

If you work out the area where it's half the base times the height of you, remember formula for an area of a triangle, which will be 30 as well.

So they got the same perimeter.

They've got the same area.

That's what these five triangles have in common.

So if I do the next one, you could see this one's a right angle triangle as well.

Perimeters 24 area is 24 that's a look of these.

Next three.

This one perimeter.

What is the perimeter 36 on?

You won't be surprised.

Areas 36 as well.

These ones that right?

And so these?

Yeah, these last three and drawing out are not right angle triangles.

We look at the perimeter or perimeter 42 area for two.

You everyone's favorites on the last one is really long.

One then.

So the perimeter is 60.

Area is 60 right?

They do remind me a perfect numbers.

The way that they kind of refer back to itself a perfect numbers when you look at the factors of the number and it adds up to itself and it reminds me of that the perimeter and the area of the same on, even though there are an infinity off possible triangles we could have with inter decides into your areas.

These are the only five that have the same perimeter in the same area, so there is five and five only.

Presumably you can upsides the same fire that I's a good point.

But the problem with that is, if we talk, say this triangle, if we double it, then we would double the perimeter and the area gets multiplied by four.

So when you double the perimeter areas mortified by four so even coat, including scaled up versions of triangles at this is the five the only five we confined.

So let me give it, give you some facts about this.

These things go back thousands of years.

It's just one of those things mathematicians study.

So these are facts about the hero nian triangles, which means they have into decides and into your area.

So they are named after a mathematician from way back 2000 years ago, hero of Alexandria.

Another hard to find.

We saw that we can do it with a Pythagorean triangle.

Pythagorean triples.

That's a well known thing.

And there are infinitely many of these on their quite easy to make.

I mean, the simplest is the 345 triangle Perimeter 12 on the area, half the base times the height is six.

So you can do this.

But any Pythagorean triangle would work as another example 17 15 8 would work as well.

So he's got an interview Perimeter, which I think is 40 on the area is, ah, interview as well.

60.

That's gonna be guaranteed to be in interview.

So they're heroes, but not superhero guaranteed.

Every pie factory and triangle is going to have this property because you're gonna have an integer perimeter on.

Yet you're guaranteed to have into your area as well.

I'm afraid what?

What I mean is your guarantee, cause it's half the base times the height.

It means you are a guarantee that one of the's sides is an even the sound of all those infinite Pythagorean triangles.

Only two of them have this property where the area is the same as the perimeter on then the other three are not right angle triangle.

So what else could we have?

S o.

I could tell you straight away.

If you wanted an equilateral triangle like this recites all the same length, whatever that is, I'm afraid that's never going tohave an integer area.

So you can't do it without collateral triangles.

They don't exist.

Wells, Could we do water about a nice sauce?

Elise Triangle.

Yes, we can do that.

I'll show you an example.

A 55 and a six.

Perimeter ISS 16 on the area is going to be 12.

So we've got injured your perimeter into jet area.

It's a bit of a cheat, though.

I don't have you noticed it was a bit of a cheat because if I drop a vertical line here, what it is is actually too right Angle triangle stuck together.

It's actually a 543 right angle triangle stuck together.

So I'm afraid Yes, you can do I saw city strangles although they are a bit of a cheat, So can we make triangles that are not a bit of a cheat?

Okay, so we'll get will go a bit more, General.

Now here's another triangle.

It's going to be a 15 2025.

So what's it gonna be?

Perimeter is 60 area is 150.

So this works.

This is one of our hero triangles, but I've cheated again because you can drop a vertical line.

You can split into two right angle triangles.

So the question is, Are they?

Are they all cheats?

Are there any triangles that aren't just made out of Pythagorean like angle triangles?

And so you can do triangles that aren't cheats?

Here is a 5 29 and 30.

So here's a triangle with a perimeter off 64 on an area of 72 on that cannot be broken up into two right angle triangles.

That a pie factory and you drop her vertical line here where we know what.

It's half the base times the height.

We're trying to work out the height now, so area is half base times height.

I'm trying to work out the height, so the height is going to be, well twice the area divided by my base, which is 144 divided by 30.

But it's not an integer, which means it's not going to be a right angle triangle with into decide When you get these.

These are called the indie compose Herbal ones.

There once that you can't break up, you break them up into right angle triangles on.

They're going to be rational sided, right angle triangles, but not always interview one.

So these, like in decompose herbal.

So why these called hero triangles named after hero of Alexandria, who mathematician 2000 years ago worked in the Library of Alexandria.

Like this big famous thing on there is a formula for the area of a triangle that is credited to hero of Alexandria, which I wasn't taught at school.

I wish I was.

I was taught area of a triangle is half the base times the height, and I've always used that.

And sometimes the height is difficult to work out, and you have to use a bit.

Pythagoras is there room work out the height and then you can do half the base times the height.

Well, here's a formula where we don't need to know the height of your triangle.

The area of this triangle is going to use a B and C Lovely.

Now the perimeter is a plus B plus C.

I'm gonna use something called a semi perimeter, which you might be able to guess it's half the perimeter and we're gonna use that in the formula for area.

The area of this triangle is the semi perimeter multiplied by semi perimeter minus a side, eh?

Multiplied by semi perimeter minus be multiplied by semi perimeter minus C on the final things.

All square root kids.

So I want to know if I wanted to work out my previous examples, which was a 5 29 30 and I don't want to work out the height.

Then I can use this formula.

It's gonna be half the perimeter multiplied by 32 minus as do Side A's five.

So that's going to be 27 multiplied by 32 minus 29.

Which is it doesn't easy three minus two minus 30.

Which is to the truth that is going to be square rooted so you can work it out.

Okay, multiplies out.

It is 72.

It's a lovely formula, isn't it?

You get the formula by Pythagoras is therapy use Pythagoras is there, um, over and over again.

Rewrite it in terms off a B and C but it turns out really meet.

It was lovely meat for you without having to work out the height of the triangle, but we were interested in triangles that have the same perimeter in the same area.

Thes air, the superheroes.

We've got a formula for area.

So let's do it right.

We got area.

Is this?

We wanted to be the same as two s.

That's the perimeter is equal to area, which is the square root thing.

Oh, well, let's just tied it up.

You know, square both sides for s squared.

Is this stuff on dhe, then weaken Divide by S s on both sides.

So we get four s equals s my a s, my speed.

That's my sea little formula on dhe.

I might not go through the whole thing, but you can do this.

I recommend if you want to solve this now and find those triangles, let's call these three terms X, y and Zed.

The letter inside four times s will become four times X plus y plus a set.

So let's say X is less than why less than let's put them in order.

And there's no reason why it shouldn't be in that order.

Yeah, it doesn't take that much to work out.

Actually, Mr you do X equals one.

Stick that into this.

Solve it.

If you solved that, you're fined three solution on the three solutions are these triangles the funny side triangles here.

If you have X equals two, you get your two right angle triangles.

If X equals three, you get no new solutions.

Andi, if you're looking at a bigger numbers, you're fine.

It's impossible if you look at these it.

This equation here on the left on Site X is the smallest number.

X is smaller than why it's smaller than set left inside, which is this will be smaller than four lots off Zed plus Zed Zed.

Zed is the biggest number out of those three, which is 12 set on on the right hand side.

Ex Wives ET it's going to be bigger than X is bigger than four, so we can say is bigger than four.

Why is bigger than four as well?

So it's bigger than four times four times that Zed here because that will be 16 zed and want to solve it.

We're looking for a number smaller than 12 said bigger than 16 said at the same time, which is impossible, which means you can stop looking End of proof.

So there's only a few things you have to check on.

You find a ll the possibilities.

So we found five triangles where the perimeter is the same as the area.

Which kind of cute right?

Look.

And you could see, like, the kind of, like, perfect numbers.

His, uh here's another trying.

I wanted to show you.

It has dried out.

This has 33 34 on 65 on.

For that, it's a perimeter off.

If I'm all correct 132 on an area.

We have a formula for that now, but I'll tell you, the area 264.

So this is a triangle where the area is double the perimeter.

When I can tell you there are 18 off those on.

I can tell you if you want your area to be some multiple the perimeter, It's always a finite.

Number two is always these lovely little finite lists.

So going to be equal to the perimeter.

We have a list of five.

We wanted to be double the perimeter.

We now have this list of 18 and same for triple and times it by four and so on.

So I've got some news.

Even though these triangles have been stood for 1000 of years, we're still learning new things about them on We've discovered some new hero triangles.

S o.

I printed them out here.

You heroes.

Yeah.

Yeah, they're kind of interesting.

You can see the point where my printer ran out of ink.

Some of these triangles come in pairs.

I'll do.

Before I talk about the special ones, I'll do an example of a pair of triangles, triangles getting work.

Let's look at those.

They're great triangles.

These really valid lovely drawn triangles.

2120 and 29 on.

We've got a 17 a 28 here, 25 on If you look at the perimeters and areas because that's what we're doing the perimeter of the 1st 1 is 70.

The area is 210.