Pythagoras Theorem: Definition & Formula

Geometry is fun!

Right?

Okay, while you ponder that over, we think geometry is one of those subjects that make maths time so much more fun and interesting! What with their head-scratching diagrams and theorems, you are always left trying to figure out the most logical and diagrammatical way to reach that conclusion.

And the Pythagoras theorem is just one of those famous gripping head-scratchers!

With a rich history that dates back to 2500 BCE, it is believed that Pythagoras, the Greek philosopher whose name forms the first half of this theorem, studied all about it when he was a student in Egypt.

Should we make it more interesting? Well, the fun fact is that the Egyptians are believed to have known all about this fascinating theorem a thousand years before Pythagoras made it famous! Kudos to the Egyptians for being one of the most scientific and advanced civilizations of their time.

Anyway, cut to the chase, let us learn all about this fascinating theorem, shall we? Also known as Pythagoras law or Pythagorean theorem, there is a lot of logic and calculations in its framework. So let us dive straight in!

Also Read: What Is Tessellation In Geometry

Pythagoras theorem: Decoding what it is all about

Okay, so to establish what exactly happens in a Pythagoras theorem, let us get the basics straight.

You have a right angled triangle in Pythagoras law.
The Pythagoras triangle is made up of three sides, vis a vis, the base (no explanation needed there), the altitude (which is the right angle side and is also known as the height or the perpendicular) and the hypotenuse (which is the longest side of the triangle).
Also, remember that a right-angled triangle is one with a 90-degree angle. Never forget that. That angle is always going to be straight as an arrow.
Now for the labels because these will form the foundation of the equation of this law. So you label the right angle side of the triangle as “a”, the base as “b” and the hypotenuse as “c”.

So far, so clear? You are with us? Okay, good. Now let us get to the formula for Pythagoras theorem.

Pythagoras theorem equation: Figuring it all out

So this theorem, in the simplest of terms, is all about figuring out one simple thing: the relationship between all the sides of this right-angled triangle. That is it. That is the core. The crux of it.

But the question is how? How do we crack the formula for Pythagoras theorem?

For that, we have to look at its core statement, which goes something like this:

The square of the hypotenuse = square of the base + square of the altitude.

To spell it out:

c² = a² + b²

Now remember, this holds true for any right angled triangle, okay? Burn that in your memory.

The Pythagoras theorem equation will always be c² = a² + b^2.And it will always conclusively prove that the square of the hypotenuse equals the sum of the square of the base and the square of the altitude.

Pythagoras theorem proof: A brief how-to

Now let us come to the meat of the matter. How do we prove this theorem in geometry? Is there a way?

Yes and yes!

So there is a method called “similar triangles”, which you can use to establish the very truth (tried and tested) of the Pythagoras theorem and it goes something like this:

NOTE: This way works on a sort of relational analysis. So say you have two triangles with corresponding angles that measure the same and have corresponding sides of the same ratio, you can call them “similar”. Thus, you can sine law to prove c² = a² + b^2.

Step 1: Let us take a right-angled triangle. Let us call it ABC. The right angle in this triangle is B. Clear?
Step 2: Cool. Now let us draw a perpendicular from B and call it BD. This perpendicular will meet AC at D, ergo BD.
Step 3: Now you have two triangles △ABD and △ACB. What is common between them? For starters, ∠A is common to both. Also, ∠ADB = ∠ABC because both of them are right angles.
Step 4: So you can effectively conclude that △ABD ∼ △ACB (because of the AA similarity).
Step 5: Now you can also prove △BCD ∼ △ACB. How? Because we have already proven △ABD ∼ △ACB, you can assume that AD/AB = AB/AC. By that logic, you can also assume AD × AC = AB².
Step 6: Making sense so far? Good. Now similarly, you know that △BCD ∼ △ACB. Now going by that logic, you can safely assume that CD/BC = BC/AC and CD × AC = BC².
Step 7: Now what do you do? Any guesses? Now you add the two equations you have. Which is AB² + BC²= (AD × AC) + (CD × AC).
Step 8: And now we are finally at the conclusion because by the above logic, AB² + BC² = AC (AD + CD). So AB² + BC²= AC². Proven! You did it! You now know the Pythagoras triangle well!

Pythagoras law: Real world applications

Phew! That was complicated, right? We totally get it if you are wondering whether the Pythagoras theorem will ever come in handy! But the truth is, it will. It is used in a lot of different fields. Let us check out some of them where the Pythagoras theorem equation finds its application:

In the field of construction and engineering

Did you know that architects actually frequently use this theorem? It helps them find what are known as unknown dimensions. It actually makes their jobs a lot simpler as they can measure unknown lengths or breadths pretty easily with Pythagoras theorem. Even engineers use it very commonly.

In face recognition systems of security cams

So here is the mind-blower! Each time you look into a security camera at someone’s door, the camera uses the Pythagoras theorem to calculate the distance between the camera and you. And then it processes that data to the lens to capture security footage. Fascinating!

In interior designing and woodworking

Woodworkers and interior designers often use this theorem to find the natural balance of the different elements at play in designing objects.

In navigation

And finally, sailors will often use the Pythagoras theorem to measure the shortest distance in order to find the fastest routes to their destination.

That was a fascinating deep dive into Pythagoras theorem, and we hope you found it just as fun and useful. That said, shall we tell you a secret? You can learn this and a lot more at EuroSchool because our faculty has what it takes to teach you the most complex topics in the simplest and most interesting manner! Visit us to find out more.