Pythagorean Theorem Explained with Examples

What Is the Pythagorean Theorem?

The Pythagorean theorem is one of the most fundamental and useful theorems in mathematics. It describes the relationship between the three sides of a right triangle:

In any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

This theorem is named after the ancient Greek mathematician Pythagoras, who is credited with its proof around 500 BCE, though the relationship was known to Babylonian mathematicians even earlier.

Understanding Right Triangles

Before using the theorem, let's make sure we understand right triangles:

• A right triangle has one angle that measures exactly 90° (a right angle)
• The side opposite the right angle is called the hypotenuse — it's always the longest side
• The other two sides are called legs (sometimes labeled a and b)

The key thing to remember: the hypotenuse is always opposite the right angle and is always the longest side.

The Formula: a² + b² = c²

The Pythagorean theorem is expressed as:

a² + b² = c²

Where:

• a and b are the lengths of the two legs
• c is the length of the hypotenuse

This means if you know any two sides of a right triangle, you can always find the third side.

Finding the Hypotenuse

Example: A right triangle has legs of length 3 and 4. Find the hypotenuse.

Step 1: Write the formula: a² + b² = c²

Step 2: Plug in the known values: 3² + 4² = c²

Step 3: Calculate: 9 + 16 = c² → 25 = c²

Step 4: Take the square root: c = √25 = 5

The hypotenuse is 5 units long.

Another example: Legs are 5 and 12.

5² + 12² = c² 25 + 144 = c² 169 = c² c = √169 = 13

Finding a Missing Leg

Example: The hypotenuse is 10 and one leg is 6. Find the other leg.

Step 1: Write the formula: a² + b² = c²

Step 2: Plug in known values: 6² + b² = 10²

Step 3: Calculate: 36 + b² = 100

Step 4: Solve for b²: b² = 100 - 36 = 64

Step 5: Take the square root: b = √64 = 8

The missing leg is 8 units long.

Notice that when finding a leg, you subtract the squares instead of adding them.

Real-World Examples

The Pythagorean theorem isn't just a classroom exercise. It appears everywhere in real life:

1. Ladder against a wall A ladder is 13 feet long and its base is 5 feet from the wall. How high up the wall does it reach?

5² + h² = 13² 25 + h² = 169 h² = 144 h = 12 feet

2. Walking diagonally You need to walk from one corner of a rectangular park to the opposite corner. The park is 300 meters long and 400 meters wide. How far is the diagonal path?

300² + 400² = d² 90,000 + 160,000 = d² d = √250,000 = 500 meters

Compared to walking along two sides (700m), the diagonal saves you 200m!

3. TV screen size TV screens are measured diagonally. If a TV is 48 inches wide and 36 inches tall, what's the screen size?

48² + 36² = d² 2,304 + 1,296 = d² d = √3,600 = 60 inches

Pythagorean Triples

Some right triangles have sides that are all whole numbers. These special sets are called Pythagorean triples:

| a | b | c | |---|---|---| | 3 | 4 | 5 | | 5 | 12 | 13 | | 8 | 15 | 17 | | 7 | 24 | 25 |

Useful trick: Any multiple of a Pythagorean triple is also a triple. For example, since 3-4-5 works, so does 6-8-10, 9-12-15, and 30-40-50.

Memorizing the common triples can save you time on exams!

Common Mistakes

1. Using the formula on non-right triangles: The Pythagorean theorem ONLY works for right triangles
2. Confusing legs and hypotenuse: c must always be the hypotenuse (longest side)
3. Forgetting to take the square root: Don't stop at c² — you need c
4. Adding instead of subtracting when finding a leg: When solving for a leg, rearrange to b² = c² - a²
5. Not checking if your answer makes sense: The hypotenuse must be longer than either leg

Summary

The Pythagorean theorem (a² + b² = c²) relates the three sides of any right triangle. To find the hypotenuse, add the squares of the legs and take the square root. To find a missing leg, subtract and take the square root. This theorem has countless real-world applications and is one of the most important tools in geometry.