There is one fact about right-angled triangles so useful that builders, sailors and astronomers have leaned on it for thousands of years. Once you see why it’s true, you’ll never forget it.
The big idea
A right triangle is any triangle with one square corner — a 90° angle. The two short sides that form that corner are called the legs, and the long side opposite it is the hypotenuse.
The Pythagorean theorem says these three lengths are locked together by one simple rule:
In words The square built on the hypotenuse has exactly the same area as the two squares built on the legs, added together.
Seeing why it’s true
Don’t just trust the formula — look at it. Draw an actual square on each side of the triangle. The area of the big square (on side c) really does equal the two smaller squares (on sides a and b) combined.
Worked example 1 — finding the hypotenuse
- Start from the theorem: \( a^2 + b^2 = c^2 \).
- Substitute the legs: \( 3^2 + 4^2 = c^2 \).
- Work out the squares: \( 9 + 16 = c^2 \), so \( c^2 = 25 \).
- Take the square root: \( c = \sqrt{25} = 5 \).
The 3–4–5 triangle This exact set of whole numbers turns up everywhere — carpenters still use it to check that a corner is perfectly square.
Worked example 2 — finding a missing leg
- Write the theorem with the unknown leg as \( b \): \( 5^2 + b^2 = 13^2 \).
- Square the known values: \( 25 + b^2 = 169 \).
- Subtract to isolate \( b^2 \): \( b^2 = 169 - 25 = 144 \).
- Take the square root: \( b = \sqrt{144} = 12 \).
When you already know the hypotenuse, you subtract instead of add — the longest side is the one being broken apart.
Where you’ll meet it
Beyond the classroom The straight-line distance between two points on a map, the diagonal of a TV screen, whether a ladder will reach a window — all of them are just the Pythagorean theorem wearing different clothes.
Practice
Try each one yourself, then reveal the full solution.
1. A right triangle has legs of 6 and 8. Find the hypotenuse.
2. The hypotenuse is 17 and one leg is 8. Find the other leg.
3. A 10 m ladder rests against a wall with its base 6 m out. How high up the wall does it reach?
You did it. That’s a full EastMath lesson — intuition, a proof you can see, worked examples, and practice that checks itself. Every lesson on the path works exactly like this.