Pythagorean Theorem: Right Triangles

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Where Did the Pythagorean Theorem Come From?

Have you ever wondered how builders make sure a corner is perfectly square? Or how your phone's GPS figures out how far away something is? The answer goes back thousands of years to a simple rule about right triangles (triangles that have one 90° angle). People all over the ancient world noticed a special pattern in these triangles.

Long before anyone wrote it as a formula, ancient civilizations used this pattern to build pyramids, lay out farms, and navigate the seas. Let's look at how this idea developed over time.

~1800 BCE

Babylonian Clay Tablets

Ancient Babylonians carved number patterns onto clay tablets. These show they already knew the relationship between the sides of a right triangle — over 1,000 years before Pythagoras!

~1100 BCE

Ancient Chinese Mathematics

Chinese mathematicians described the same rule in a book called the Zhoubi Suanjing. They used it to measure distances and build structures.

~530 BCE

Pythagoras & the Greeks

The Greek mathematician Pythagoras (or his followers) wrote the first known proof of the rule. That's why we call it the Pythagorean Theorem today.

~300 BCE

Euclid's Elements

The Greek scholar Euclid included a famous proof of the theorem in his textbook, Elements. This book was used for over 2,000 years!

Today

Everywhere Around Us

Engineers, architects, game designers, and scientists use the Pythagorean Theorem every single day. It is one of the most useful ideas in all of math.

So here is the big question this lesson answers: If you know two sides of a right triangle, how do you find the third side? The Pythagorean Theorem gives you the tool to do exactly that.

SECTION 2

Core Principles & Definitions

Before we jump into the formula, let's make sure we understand the key vocabulary. These are the building blocks you need.

Right Triangle

A triangle that has exactly one right angle (a 90° angle). You can spot it by the small square drawn in the corner.

Legs (a and b)

The two shorter sides that form the right angle. We label them a and b. It doesn't matter which leg is a and which is b.

Hypotenuse (c)

The longest side of a right triangle. It is always across from the right angle — never touching it. We label it c.

Squaring a Number

When you square a number, you multiply it by itself. For example, 5² = 5 × 5 = 25.

Square Root

The square root is the opposite of squaring. √25 = 5 because 5 × 5 = 25. You use it to "undo" a square.

✦ KEY TAKEAWAY

Think of a right triangle like a book propped open on a table. The two edges touching the table are the legs, and the slanted cover stretching between them is the hypotenuse. The hypotenuse is always the longest side because it stretches across the widest part of the triangle.

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Seeing the Theorem in Action

The best way to understand the Pythagorean Theorem is to see it. The diagram below shows a right triangle with a square built on each of its three sides. Notice something amazing: the areas of the two smaller squares add up to the area of the big square. That is the Pythagorean Theorem!

A 3-4-5 right triangle with squares drawn on each side. The cyan square (area 16) plus the violet square (area 9) equals the pink square (area 25). That's the Pythagorean Theorem!

In this diagram, the legs are 3 and 4 units long, and the hypotenuse is 5 units long. When you square each side, you get 9, 16, and 25. Check it: 9 + 16 = 25. It works! This is true for every right triangle ever, no matter how big or small.

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The Formula & How to Use It

Now let's write the rule as a formula you can use to solve problems. The Pythagorean Theorem says:

PYTHAGOREAN THEOREM

a² + b² = c²

a and b = the two legs (shorter sides touching the right angle). c = the hypotenuse (the longest side, across from the right angle).

You will run into two kinds of problems. Sometimes you need to find the hypotenuse. Other times you need to find a missing leg. Each one uses the same formula, just rearranged a little.

FINDING THE HYPOTENUSE

c = √(a² + b²)

Square both legs, add them together, then take the square root. This gives you the length of the hypotenuse.

FINDING A MISSING LEG

a = √(c² − b²)

Square the hypotenuse, subtract the square of the known leg, then take the square root. This gives you the missing leg.

💡 Quick Tip

Always identify which side is the hypotenuse first. It is the side across from the 90° angle. If the problem asks you to find the hypotenuse, you add the squared legs. If it asks you to find a leg, you subtract.

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Two Types of Problems You'll See

Every Pythagorean Theorem problem boils down to one of two types. The diagram below shows both side by side so you can see how the formula changes depending on what is missing.

Type 1 (left): You know both legs and need the hypotenuse — add then square root. Type 2 (right): You know the hypotenuse and one leg — subtract then square root.

Quick reference: choosing the right formula

What You Know	What You Need	Formula to Use
Both legs (a and b)	Hypotenuse (c)	c = √(a² + b²)
Hypotenuse (c) and one leg	The other leg	a = √(c² − b²)

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Worked Example: Step by Step

Let's walk through a full problem together. Read each step carefully before moving to the next one.

📐 Problem

A ladder leans against a wall. The bottom of the ladder is 6 feet from the wall, and the ladder is 10 feet long. How high up the wall does the ladder reach?

Finding the Missing Leg

Step 1 — Draw It OutPicture the wall, the ground, and the ladder forming a right triangle. The wall and ground meet at a 90° angle. The ladder is the hypotenuse (c = 10) because it's across from the right angle. The distance from the wall is a leg (b = 6). The height on the wall is the missing leg (a = ?).

Step 2 — Write the FormulaSince we are finding a missing leg, we use: a² + b² = c², which we rearrange to a² = c² − b².

a² = c² − b²

Step 3 — Plug In the NumbersSubstitute the values we know. The hypotenuse c = 10 and the known leg b = 6.

a² = 10² − 6² = 100 − 36 = 64

Step 4 — Take the Square RootFind the square root of 64. Ask yourself: what number times itself gives 64? That's 8, because 8 × 8 = 64.

a = √64 = 8 feet

Step 5 — Check Your AnswerPlug all three sides back into a² + b² = c² to verify. 8² + 6² = 64 + 36 = 100. And 10² = 100. ✓ It checks out! The ladder reaches 8 feet up the wall.

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Common Mistakes & Helpful Tips

The Pythagorean Theorem is straightforward, but there are a few traps students fall into. Let's go over them so you can avoid them.

Watch out for these common errors!

Common Mistake	Why It's Wrong	What To Do Instead
Using the formula on any triangle	The theorem only works for right triangles — triangles with a 90° angle.	Always check for the right-angle symbol (the little square) before using a² + b² = c².
Mixing up legs and hypotenuse	If you put the hypotenuse value in for a leg, your answer will be wrong.	Remember: c is always the longest side and is opposite the right angle.
Adding instead of subtracting when finding a leg	When finding a leg, you need to subtract, not add.	Finding hypotenuse → add. Finding a leg → subtract. Then take the square root.
Forgetting the square root at the end	You'd have the square of the side, not the actual side length.	After adding or subtracting, always take the square root to get the actual length.

🎯 REMEMBER THIS

Think of the Pythagorean Theorem like a recipe. Before you start cooking, you check which ingredients you already have. Identify your known sides first, then decide whether to add or subtract. The hypotenuse is always the "biggest ingredient" — it never gets subtracted from the other sides.

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Where Does This Lead Next?

The Pythagorean Theorem is one of the most important stepping stones in math. Once you master it, you will use it in many future topics. Here is a preview of what's ahead.

How today's lesson connects to future math

What You Learn Now	What You'll Learn Later
Finding sides of right triangles with whole-number answers	Working with answers that are irrational numbers (like √2 ≈ 1.414)
Flat (2D) right triangles	Using the theorem in 3D space to find distances in boxes and rooms
a² + b² = c² with numbers	The distance formula in coordinate geometry: d = √((x₂ − x₁)² + (y₂ − y₁)²)
Right triangles only	Trigonometry (sine, cosine, tangent) — finding angles and sides in all kinds of triangles

One exciting connection is the distance formula. When you plot two points on a graph, you can draw a right triangle between them. The horizontal distance is one leg, the vertical distance is the other leg, and the straight-line distance between the points is the hypotenuse. So the distance formula is just the Pythagorean Theorem in disguise!

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Practice Problems

Time to test yourself! Try each problem on your own before looking at the answer. The problems start easy and get harder as you go.

PROBLEM 1 — CONCEPTUAL

In a right triangle, one side is called the hypotenuse. How can you always identify which side is the hypotenuse?

PROBLEM 2 — BASIC CALCULATION

A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?

PROBLEM 3 — INTERMEDIATE

A right triangle has a hypotenuse of 15 and one leg of 9. Find the length of the other leg.

PROBLEM 4 — APPLIED

A baseball diamond is a square with sides of 90 feet. A catcher at home plate throws the ball to second base. How far is the throw? (Hint: the diagonal of a square cuts it into two right triangles.)

PROBLEM 5 — CRITICAL THINKING

Can a triangle with sides 7, 10, and 12 be a right triangle? Use the Pythagorean Theorem to explain your answer.

SUMMARY

Lesson Summary

The Pythagorean Theorem states that in any right triangle, the sum of the squares of the two legs (a and b) equals the square of the hypotenuse (c). Written as a formula: a² + b² = c². The hypotenuse is always the longest side and sits across from the 90° angle.

To find the hypotenuse, square both legs, add them, and take the square root. To find a missing leg, square the hypotenuse, subtract the square of the known leg, and take the square root. Always identify the hypotenuse first, and remember: this formula only works for right triangles. You can also use the theorem in reverse to test whether a triangle is a right triangle.