Mathematics · Lesson 05
Pythagoras & Triangles
Over 2,500 years ago, Pythagoras proved that a simple relationship connects the three sides of every right triangle. That one idea is still used by architects, engineers, GPS systems, and game developers today.
Who was Pythagoras?
Pythagoras of Samos was a Greek philosopher and mathematician who lived around 570–495 BCE. He founded a school of thought that believed mathematics was the foundation of all reality — long before anyone had calculators, computers, or even algebra as we know it today.
Although the theorem that bears his name was likely known to Babylonian and Indian mathematicians even earlier, Pythagoras and his followers are credited with providing the first formal proof — a logical argument showing it must always be true, for every right triangle, without exception.
The Pythagorean Theorem
A right triangle is any triangle with one 90° angle (the right angle, shown with a small square). The two sides that form the right angle are called the legs. The side opposite the right angle — always the longest side — is the hypotenuse.
c is the hypotenuse.
The theorem states: the square of the hypotenuse equals the sum of the squares of the two legs. Square leg a, square leg b, add them together — and you get the square of hypotenuse c.
Solving for the hypotenuse
If you know both legs, find c by taking the square root of (a² + b²):
Famous example — the 3-4-5 triangle
The simplest whole-number right triangle has sides 3, 4, and 5. Check: 3² + 4² = 9 + 16 = 25 = 5². Ancient Egyptian rope-stretchers used a rope with 12 equally-spaced knots tied in a 3-4-5 triangle to create perfect right angles when building the pyramids.
Pythagoras Blueprint Lab
Move from “memorize the formula” to “watch the areas behave.” The blue and green leg-squares are physical containers, the orange hypotenuse square is the combined destination, and the angle readouts show how this theorem quietly leads toward trigonometry.
Measurements
Water Proof
These bars track how the blue and green square areas “pour” into the orange square. The total fill always lands exactly on c².
Ramp Builder Challenge
Treat a as the rise and b as the run. The hypotenuse becomes the actual ramp or roof truss length.
Types of Triangles
Every triangle has three angles that add up to exactly 180°. The relationship between those angles determines which of the three types it is — and whether Pythagoras applies.
How to Solve Problems
Finding the hypotenuse (c)
- 1Identify the two legs: a and b.
- 2Square each leg: calculate a² and b².
- 3Add them: a² + b².
- 4Take the square root of the result: c = √(a² + b²).
Finding a missing leg (a or b)
- 1Rearrange: if you know c and b, then a² = c² − b².
- 2Calculate c² and b².
- 3Subtract: c² − b².
- 4Take the square root: a = √(c² − b²).
Real-World Applications
The Pythagorean theorem isn't just a classroom exercise — it's one of the most-used mathematical tools in the world.
Practice Problems
Type your answer and press Check. Round decimals to one place.
1. A right triangle has legs a = 3 and b = 4. What is the hypotenuse c?
2. A right triangle has legs a = 5 and b = 12. What is the hypotenuse c?
3. A right triangle has hypotenuse c = 10 and one leg b = 6. What is leg a?
4. A screen is 9 inches wide and 12 inches tall. How long is the diagonal (corner to corner)?
5. A ladder 13 feet long leans against a wall. Its base is 5 feet from the wall. How high up the wall does the ladder reach?