What is Pythagoras theorem?
A right-angled triangle has one angle of exactly 90°. The side opposite this angle is the longest side and is called the hypotenuse (c). The other two sides (a and b) are called the legs.
Pythagoras theorem says that if you build a square on each side, the area of the big square on the hypotenuse is exactly equal to the areas of the two smaller squares added together.
Why does it work? A visual proof
Take four copies of the same right triangle and arrange them inside a large square of side (a + b). The empty space in the middle is a tilted square of side c, so its area is c².
Now slide the same four triangles into a different arrangement inside the same large square. This time the empty space is two squares: one of area a² and one of area b². The large square and the four triangles did not change, so the empty space must be the same: a² + b² = c². Press play on the animation to watch the squares fill up.
Converse of Pythagoras theorem
The theorem also works backwards. If the sides of a triangle satisfy a² + b² = c², the triangle must have a right angle opposite side c. Builders use this with a 3-4-5 rope to mark perfect corners.
Pythagorean triples
Whole numbers that fit the rule are called Pythagorean triples. Common ones are 3-4-5, 5-12-13, 8-15-17 and 7-24-25. Any multiple of a triple (6-8-10, 9-12-15) is also a triple.
Key formulas and definitions
- a² + b² = c² (c is the hypotenuse)
- c = √(a² + b²)
- a = √(c² − b²)
- Hypotenuse = side opposite the 90° angle, always the longest side
Worked examples
1. The two shorter sides of a right triangle are 6 cm and 8 cm. Find the hypotenuse.
c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.
2. A 13 m ladder leans against a wall. Its foot is 5 m from the wall. How high up the wall does it reach?
The ladder is the hypotenuse. h² = 13² − 5² = 169 − 25 = 144, so h = 12 m.
3. Is a triangle with sides 7, 10 and 12 right-angled?
Check the longest side: 7² + 10² = 49 + 100 = 149, but 12² = 144. They are not equal, so it is not a right triangle.
Common mistakes
- Adding the hypotenuse as one of the legs. Always put the longest side alone on one side: c² = a² + b².
- Forgetting the square root at the end: c² = 100 means c = 10, not 100.
- Using the theorem on a triangle that has no right angle.
- Writing √(a² + b²) = a + b. Square roots cannot be split over addition.