The Pythagorean Theorem
What the theorem says, when it applies, and how to use it to find missing sides.
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:
a² + b² = c²
where c is the hypotenuse, the side opposite the right angle and the longest side.
When to use it
Only for right-angled triangles. If a triangle has no right angle, you need other tools such as the sine or cosine rule.
Finding the hypotenuse
Example. The shorter sides are 6 and 8.
- c² = 6² + 8² = 36 + 64 = 100.
- c = √100 = 10.
Finding a shorter side
Example. The hypotenuse is 13 and one side is 5.
- 5² + b² = 13², so 25 + b² = 169.
- b² = 144.
- b = 12.
Pythagorean triples
Sets of whole numbers that satisfy the theorem include 3, 4, 5; 5, 12, 13; and 8, 15, 17. Multiples of these, such as 6, 8, 10, also work.
Checking whether a triangle is right-angled
Sides 7, 24, 25: 7² + 24² = 49 + 576 = 625 = 25². Yes, it is a right-angled triangle.
Common mistakes
- Adding the wrong sides. The hypotenuse is on one side of the equation alone.
- Forgetting to take the square root at the end.
- Using it for non-right triangles.
Quick recap
a² + b² = c² for right triangles. Identify the hypotenuse first, substitute, and take the square root.