Pythagorean Theorem: Complete Guide, Examples and Practice

What Is the Pythagorean Theorem?

The Pythagorean theorem is a mathematical rule that holds for right triangles. It is named after the Greek philosopher and mathematician Pythagoras (c. 570–495 BCE), although the relationship was already used by the Babylonians and Egyptians long before him.

The theorem states: in any right triangle, the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides.

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The Pythagorean Formula

The Pythagorean theorem is written as:

Where:

• = length of the first leg of the right triangle
• = length of the second leg of the right triangle
• = length of the hypotenuse (the longest side, opposite the right angle)

From this base formula we can derive three variations:

Finding the hypotenuse (c):

Finding side a:

Finding side b:

💡 Easy way to remember: "The square of the hypotenuse equals the sum of the squares of the other two sides." The hypotenuse is always the longest side and always opposite the 90° angle.

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How to Use the Pythagorean Formula

Example 1: Finding the Hypotenuse ⭐

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

Step 1: Write down the formula

Step 2: Substitute the values

Step 3: Take the square root of both sides

✅ Answer: hypotenuse = 5 cm

Example 2: Finding One of the Legs ⭐⭐

A ladder of length m leans against a wall. The foot of the ladder is m away from the wall. How high up the wall does the ladder reach?

Step 1: Use the formula for finding side a

Step 2: Substitute the values

Step 3: Calculate

✅ Answer: the ladder reaches 12 m up the wall

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Example 3: Common Mistake — Misidentifying the Hypotenuse ⭐

The most common mistake: students do not know which side is the hypotenuse and which sides are the legs.

Problem: A triangle has sides 6 cm, 8 cm and 10 cm. Is it a right triangle? Which side is the hypotenuse?

Wrong approach ❌: A student computes and concludes "it is not a right triangle".

The student picked the wrong side as the hypotenuse — the hypotenuse is always the longest side (c = 10, not 8).

Correct approach ✅: Use the longest side as c:

✅ This is a right triangle. The hypotenuse = 10 cm (and this is the 6-8-10 triple, a multiple of 3-4-5)

💡 Remember: The hypotenuse (c) is always the longest side and lies opposite the 90° angle. If you are unsure, use the longest side as c.

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Pythagorean Triples Worth Memorising

A Pythagorean triple is a set of three positive integers that satisfy . Memorising these can save a lot of time in exams.

💡 Multiples pattern: If is a Pythagorean triple, so are its multiples: , , , and so on.

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Worked Examples + Solutions

Problem 1 ⭐: A rectangular field is 8 m long and 6 m wide. What is the length of its diagonal?

The diagonal of a rectangle forms a right triangle with the length and width as its legs.

✅ Diagonal length = 10 m

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Problem 2 ⭐⭐: Two people set off from the same point. The first walks 9 km east, the second walks 12 km north. How far apart are they now?

East and north meet at a right angle, so we use the Pythagorean theorem.

✅ The distance between them = 15 km

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Problem 3 ⭐⭐: Is a triangle with sides 5 cm, 7 cm and 9 cm a right triangle?

Check whether holds (c is the longest side = 9):

Since , this triangle is not a right triangle.

Learn more about how to find the area of a triangle — a concept closely related to the Pythagorean theorem.

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Practice Problems

⭐ Problem 1: A right triangle has cm and cm. Find .

⭐⭐ Problem 2: A ladder's hypotenuse m, and its distance from the wall is m. How high up the wall does it reach?

⭐⭐ Problem 3: Is a Pythagorean triple? Prove it.

⭐⭐⭐ Problem 4: A ship's sail is a right triangle. It is 9 m tall and its base is 12 m. How long is the rope from the top to the end of the base?

Answer key:

1. 13 cm. This is the 5-12-13 Pythagorean triple — worth memorising.
2. 8 m. This is the 6-8-10 triple (a multiple of 3-4-5).
3. ✓ Yes, it is a Pythagorean triple — twice (3, 4, 5).
4. 15 m. The 9-12-15 triple (three times 3-4-5).

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Summary

1. The Pythagorean theorem holds for right triangles: , where is the hypotenuse opposite the 90° angle.
2. From this base formula you can find the hypotenuse , or a leg or .
3. Memorise the common Pythagorean triples: (3,4,5), (5,12,13), (8,15,17) — and all their multiples.
4. The most common mistake is misidentifying the hypotenuse — the hypotenuse is always the longest side and lies opposite the 90° angle.
5. The Pythagorean theorem is used in construction, navigation and how to find the area of a triangle. Also master how to calculate ratios for more advanced geometry problems.

Want to learn maths more effectively? Discover the strategies in 5 Best Approaches to Learning Maths.