Pythagoras’ Theorem & Hypotenuse Formula

Welcome to the Army Public School Babina digital learning portal. These specialized chapter notes are curated specifically to help our students master core Mathematics concepts and prepare efficiently for school and board examinations.

A Discovery That Changed the World

Imagine you’re building a treehouse and need to make sure the walls are perfectly straight. Or perhaps you’re trying to find the shortest path across a park. How would you figure out these distances?

Over 2000 years ago, an ancient Greek mathematician made a discovery that would help solve these problems forever. This discovery is called Pythagoras’ Theorem, and it’s one of the most famous and useful rules in all of mathematics.

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What is Pythagoras’ Theorem?

Pythagoras’ Theorem is a rule that applies to right-angled triangles – triangles that have one angle of exactly 90 degrees (a square corner).

The Simple Statement

Here’s what the theorem says in plain English:

“In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.”

But what does that actually mean? Let’s break it down.

The Area Way of Thinking About It

Here’s a really cool way to understand the theorem. Imagine you draw a square on each side of a right-angled triangle. Pythagoras’ Theorem tells us that:

The area of the square on the hypotenuse = the area of the square on one shorter side + the area of the square on the other shorter side

This is why the theorem is so beautiful – it’s all about areas fitting together perfectly!

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The Formula: $a^2 + b^2 = c^2$

If we label the sides of a right-angled triangle like this:

* a = one shorter side
* b = the other shorter side
* c = the hypotenuse (the longest side)

Then Pythagoras’ Theorem can be written as:

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

This is the famous formula that students all over the world learn!

What does the formula mean?

* $a^2$ means “a multiplied by a” (a squared)
* $b^2$ means “b multiplied by b” (b squared)
* $c^2$ means “c multiplied by c” (c squared)

So the formula is saying: “a squared plus b squared equals c squared”.

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Who Invented Pythagoras’ Theorem?

The Man Behind the Name

The theorem is named after Pythagoras of Samos, a Greek mathematician who lived from approximately 570 BC to 495 BC.

Pythagoras was more than just a mathematician – he was also a philosopher and the founder of a group called the Pythagorean brotherhood. He’s often described as the first “pure” mathematician.

However, here’s an interesting twist: Pythagoras might not have been the first person to discover this theorem!

The Real History

So why is it called Pythagoras’ Theorem? Tradition credits Pythagoras with being the first person to PROVE the theorem mathematically. The Babylonians and Egyptians knew it worked, but Pythagoras (or his followers) may have been the first to demonstrate WHY it always works.

A Fun Fact

Legend says that when Pythagoras discovered the proof, he was so excited that he sacrificed a bull to celebrate – even though he was generally against animal sacrifice!

Note: We don’t have any writings from Pythagoras himself because his school kept their knowledge secret and didn’t write things down. Everything we know about him comes from what others wrote about him later!

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The Hypotenuse Formula

Students often ask: “How do I find the hypotenuse?”

Since the hypotenuse is the longest side (labelled c), we can rearrange the formula to solve for it:

$$c = \sqrt{a^2 + b^2}$$

This means: “The hypotenuse equals the square root of (a squared plus b squared)”

When to use this: When you know the lengths of the two shorter sides and want to find the hypotenuse.

Finding a Shorter Side

What if you know the hypotenuse and one shorter side, but need to find the other shorter side?

You can rearrange the formula like this:

$$a = \sqrt{c^2 – b^2} \quad \text{or} \quad b = \sqrt{c^2 – a^2}$$

When to use this: When you know the hypotenuse and one shorter side, and need to find the other shorter side.

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Example 1: Finding the Hypotenuse (The Classic 3-4-5 Triangle)

Let’s start with the most famous example of all – the 3-4-5 triangle.

Problem: A right-angled triangle has shorter sides of length 3 cm and 4 cm. What is the length of the hypotenuse?

Solution:

1. Label the sides: $a = 3$, $b = 4$, $c = ?$
2. Use the formula: $a^2 + b^2 = c^2$
3. Substitute the values: $3^2 + 4^2 = c^2$
4. Calculate the squares: $9 + 16 = c^2$
5. Add: $25 = c^2$
6. Square root both sides: $c = \sqrt{25} = 5$

Answer: The hypotenuse is 5 cm

Why this is special: 3, 4, and 5 are called a Pythagorean triple – three whole numbers that fit the theorem perfectly. Other examples include 5-12-13 and 8-15-17.

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Example 2: Finding a Shorter Side

Problem: A right-angled triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. What is the length of the other shorter side?

Solution:

1. Label the sides: $a = 5$, $c = 13$, $b = ?$
2. Use the formula: $a^2 + b^2 = c^2$
3. Substitute the values: $5^2 + b^2 = 13^2$
4. Calculate the squares: $25 + b^2 = 169$
5. Subtract 25 from both sides: $b^2 = 169 – 25 = 144$
6. Square root both sides: $b = \sqrt{144} = 12$

Answer: The other shorter side is 12 cm

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Example 3: A Real-Life Problem – The Ladder

Problem: You have a ladder that is 5 metres long. You want to lean it against a wall so that the bottom of the ladder is 3 metres away from the wall. How high up the wall will the ladder reach?

Solution:

This forms a right-angled triangle where:
* The ladder is the hypotenuse: $c = 5\text{ m}$
* The distance from the wall is one shorter side: $a = 3\text{ m}$
* The height up the wall is the other shorter side: $b = ?$

1. Use the formula: $a^2 + b^2 = c^2$
2. Substitute: $3^2 + b^2 = 5^2$
3. Calculate: $9 + b^2 = 25$
4. Subtract: $b^2 = 25 – 9 = 16$
5. Square root: $b = \sqrt{16} = 4$

Answer: The ladder will reach 4 metres up the wall.

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Why Is Pythagoras’ Theorem So Useful?

Pythagoras’ Theorem isn’t just something you learn for an exam – it has real-world applications everywhere!

In Construction and Building

Builders use the 3-4-5 rule to make sure walls and corners are perfectly square (at right angles). If a triangle has sides 3, 4, and 5, the angle between the 3 and 4 sides MUST be 90 degrees!

In Navigation and Mapping

When you look at a map and want to find the shortest distance between two points, you’re using Pythagoras’ Theorem – especially if the points aren’t directly north-south or east-west of each other.

In Technology

In Everyday Life

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How to Check if a Triangle is Right-Angled

Here’s a clever trick: If you know all three sides of a triangle, you can check if it has a right angle by using Pythagoras’ Theorem backwards.

The rule: If $a^2 + b^2 = c^2$ (where c is the longest side), then the triangle is right-angled.

Example: Does a triangle with sides 8, 15, and 16 have a right angle?

* Check: $8^2 + 15^2 = 64 + 225 = 289$
* $16^2 = 256$
* Since $289 \neq 256$, then NO, it’s not right-angled

Example: Does a triangle with sides 6, 8, and 10 have a right angle?

* Check: $6^2 + 8^2 = 36 + 64 = 100$
* $10^2 = 100$
* Since $100 = 100$, then YES, it is right-angled!

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Quick Summary

What	The Rule
Pythagoras’ Theorem	In a right-angled triangle, $a^2 + b^2 = c^2$
Hypotenuse	The longest side, opposite the right angle
Finding the hypotenuse	$c = \sqrt{a^2 + b^2}$
Finding a shorter side	$a = \sqrt{c^2 – b^2}$
Who discovered it?	Named after Pythagoras (570-495 BC), but known to earlier civilisations
Pythagorean triple	Three whole numbers that fit the theorem (e.g., 3, 4, 5)

Summary

Pythagoras’ Theorem is over 2000 years old, but it’s still one of the most used mathematical rules in the world today. Every time you see a right angle, whether it’s in a building, a video game, or a map, there’s a good chance Pythagoras’ Theorem is working behind the scenes!

So the next time someone asks you what the most famous maths rule is, you can confidently say: “$a^2 + b^2 = c^2$ – and I know exactly what it means!”

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