Concept · Geometry
The Pythagorean Theorem Explained Simply
a² + b² = c² is a statement about areas, not just a formula. Once you see the squares, the theorem stops needing to be memorised.
What the theorem says
The Pythagorean theorem states that in any 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 always the side opposite the right angle.
The word 'square' is doing real work there. Build an actual square on each of the three sides, and the two smaller squares have exactly the same total area as the largest one. That is the theorem — a statement about areas that happens to be useful for finding lengths.
Finding the hypotenuse
When both legs are known, square them, add, and take the positive root.
- a² + b² = c²the Pythagorean theorem
- 6² + 8² = c²substitute the two legs
- 36 + 64 = 100square each leg and add
- c = 10take the positive root — a length cannot be negative
The last line matters. Every positive number has two square roots, one positive and one negative, but a side length is a distance, so only the positive value is meaningful.
Finding a missing leg
When the hypotenuse and one leg are known, rearrange before substituting. Subtract rather than add — the hypotenuse square is the total, and you are recovering a part of it.
- a² = c² − b²rearrange for the missing leg
- a² = 13² − 5²substitute what is known
- a² = 169 − 25 = 144evaluate
- a = 12take the positive root
Pythagorean triples worth recognising
A Pythagorean triple is a set of three whole numbers that satisfies the theorem. Recognising them saves time, because the answer comes out without a calculator.
| Triple | Check | Multiples also work |
|---|---|---|
| 3, 4, 5 | 9 + 16 = 25 | 6-8-10, 9-12-15, 30-40-50 |
| 5, 12, 13 | 25 + 144 = 169 | 10-24-26, 15-36-39 |
| 8, 15, 17 | 64 + 225 = 289 | 16-30-34 |
| 7, 24, 25 | 49 + 576 = 625 | 14-48-50 |
Any multiple of a triple is also a triple, because scaling all three sides by the same factor scales every square by the factor squared, and the equation survives.
The converse: testing for a right angle
The theorem works in reverse. If the three sides of a triangle satisfy a² + b² = c², the triangle must be right-angled. That converse is how builders check a corner is square using nothing but a tape measure.
- If a² + b² = c², the angle opposite c is exactly 90°.
- If a² + b² > c², that angle is acute — smaller than 90°.
- If a² + b² < c², that angle is obtuse — larger than 90°.
So the same three numbers tell you not only whether the triangle is right-angled, but which way it leans if it is not.
Where it leads
The distance formula on a coordinate grid is the Pythagorean theorem with the legs written as coordinate differences: the distance between two points is √((x₂ − x₁)² + (y₂ − y₁)²). Nothing new has been introduced — the horizontal and vertical gaps are the two legs.
For triangles without a right angle, the law of cosines generalises the theorem by adding a correction term. When the angle is exactly 90° that term vanishes and the law of cosines collapses back into a² + b² = c².
Questions about pythagorean theorem
Does the Pythagorean theorem work on any triangle?
No — only on right-angled ones. For other triangles you need the law of cosines, which is the same relationship with a correction term that disappears when the angle is 90°.
How do I know which side is the hypotenuse?
It is the side opposite the right angle, and it is always the longest side of the triangle. If a diagram is unclear, find the right angle first and look straight across from it.
Why are the sides squared?
Because the theorem is about the areas of squares built on each side. Squaring a length gives the area of the square drawn on it, and the theorem says the two smaller areas sum to the largest.
Why is the answer usually not a whole number?
Because most sums of two squares are not perfect squares. Triples like 3-4-5 are the exception, not the rule, so the exact answer is normally a square root.