Pythagorean Theorem- Complete Guide with Examples

What Is the Pythagorean Theorem?

The Pythagorean Theorem is a fundamental principle in geometry that describes the relationship between the three sides of a right triangle. It states that the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides.

That's it. That's the whole thing. But don't let its simplicity fool you — this theorem shows up everywhere from construction to video games.

The Formula

For a right triangle with legs a and b, and hypotenuse c:

a² + b² = c²

The hypotenuse is always the side opposite the right angle (the 90-degree angle). It's also always the longest side.

Understanding the Parts

• Legs (a and b) — the two sides that form the right angle
• Hypotenuse (c) — the side opposite the right angle, never touches the 90-degree corner
• Right angle — the 90-degree angle between the two legs

Practical Examples

Example 1: Finding the Hypotenuse

A ladder leans against a wall. The base is 3 feet from the wall, and it reaches 4 feet up. How long is the ladder?

3² + 4² = c²
9 + 16 = c²
25 = c²
c = 5 feet

The ladder is 5 feet long. Classic 3-4-5 triangle.

Example 2: Finding a Leg

A baseball diamond is a square with 90 feet between bases. What's the distance from home plate to second base (diagonally)?

90² + 90² = c²
8100 + 8100 = c²
16200 = c²
c ≈ 127.28 feet

The diagonal distance is roughly 127 feet.

Example 3: Real-World Problem

You have a 12-foot pole and need to lean it against a wall. The base can be placed 5 feet from the wall. How far up the wall will it reach?

5² + b² = 12²
25 + b² = 144
b² = 119
b ≈ 10.9 feet

The pole reaches about 10.9 feet up the wall.

Common Pythagorean Triples

These are integer sets that satisfy a² + b² = c². Memorizing them saves time:

Any multiple of these also works. 6-8-10 is just 3-4-5 doubled.

How to Use the Pythagorean Theorem

Step 1: Identify which two sides you know and which one you need to find.

Step 2: Plug the known values into a² + b² = c². If finding a leg, rearrange to a² = c² - b².

Step 3: Solve for the unknown. Take the square root at the end.

Step 4: Check your work. Does the answer make sense? Is it the longest side if it's the hypotenuse?

Common Mistakes

• Forgetting to take the square root — you solve for c², not c
• Using the wrong side as the hypotenuse — it must be opposite the right angle
• Mixing up which leg to subtract — only do c² - b² when finding the other leg
• Assuming it's a right triangle — the theorem only works if one angle is exactly 90 degrees

Applications

You encounter this theorem more than you realize:

• Construction — ensuring walls are level, calculating roof slopes
• Navigation — shortest distance between two points
• Surveying — measuring distances without crossing obstacles
• Sports — field diagonals, track layouts
• Computer graphics — calculating pixel distances, collision detection

The Converse Is Also True

Here's something most people forget: the converse works too. If three sides satisfy a² + b² = c², then the triangle must be a right triangle. This is useful for checking whether something is actually perpendicular.

Pythagorean Theorem vs. Distance Formula

The distance formula is just the Pythagorean Theorem in disguise. For two points (x₁, y₁) and (x₂, y₂):

distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

The horizontal difference and vertical difference are the legs. The direct distance is the hypotenuse.