The Pythagorean Theorem

One of Mathematics' Most Famous Results

What Is the Pythagorean Theorem?

For any right triangle, the relationship between the sides is:

• a and b are the two shorter sides (legs)
• c is the longest side (hypotenuse), opposite the right angle

Named after the Greek mathematician Pythagoras (~570-495 BC), though known to Babylonians over 1,000 years earlier.

Visual Proof

The area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides.

Real-World Applications

• Construction -- ensuring walls are square and foundations are level
• Navigation -- calculating straight-line distances between two points
• Architecture -- designing roof pitches and ramp angles
• GPS -- determining positions from satellite distances
• Computer graphics -- calculating distances between pixels

Try It Yourself

Problem: A ladder leans against a wall. The base is 6 feet from the wall, and the ladder is 10 feet long. How high up the wall does it reach?

The ladder reaches 8 feet up the wall.