The Pythagorean Theorem
a² + b² = c², the single most reused formula in all of maths.
The explanation
In a right triangle, the two short sides (legs) and the longest side (hypotenuse) are linked by a² + b² = c².
The hypotenuse is always c, and it is always opposite the right angle — the longest side.
Legs 3 and 4: 9 + 16 = 25, so c = 5.
To find a leg instead, rearrange rather than adding. With c = 13 and a = 5: 5² + b² = 13², so b² = 169 − 25 = 144, and b = 12.
Triples worth memorising, since they show up everywhere: 3-4-5, 5-12-13, 8-15-17, and multiples like 6-8-10.
The theorem holds in a right triangle and only in a right triangle; its converse states that if a² + b² = c² then the angle opposite c is right, making it a test for right angles rather than just a length formula.
The comparison version classifies any triangle: with c the longest side, a² + b² > c² means acute, = c² means right, and < c² means obtuse.
Its reach beyond triangles is what makes it central. The distance formula is the theorem applied to the horizontal and vertical gaps between two points. The equation of a circle, x² + y² = r², is the theorem stating that every point on the circle sits a fixed distance from the centre. The Pythagorean identity sin²θ + cos²θ = 1 is the same statement on the unit circle. Each of these is one theorem wearing different notation.
When the answer is not a perfect square, leave it exact as a simplified radical: √52 = 2√13 is preferable to a rounded decimal unless the context asks for one.
Worked example
A ladder 13 ft long leans against a wall with its base 5 ft out. How high does it reach?
- The ladder is the hypotenuse: c = 13, one leg a = 5.
- 5² + b² = 13² → 25 + b² = 169.
- b² = 144.
Answer: 12 ft
Common mistakes
- Adding when you should subtract: treating a leg as the hypotenuse.
- Stopping at c² and forgetting to take the square root.