The Pythagorean Theorem
The Pythagorean Theorem is one of the most recognized equations in all of mathematics:
$$a^2 + b^2 = c^2$$
It describes a relationship that holds for every right triangle, everywhere, without exception. The letters \(a\) and \(b\) represent the two shorter sides (called the legs), and \(c\) represents the hypotenuse, which is always the longest side and always sits opposite the right angle.
The theorem is named after Pythagoras, a Greek mathematician and philosopher who lived around 570–495 BC. The relationship itself was actually known in Babylon and India long before his time, but Pythagoras (or his followers, known as the Pythagoreans) are credited in the Western tradition with producing the first formal proof. The name has stuck ever since.
Finding the Hypotenuse
If you know both legs, finding the hypotenuse is a direct calculation.
Example: A right triangle has legs of length 6 and 8. Find the hypotenuse.
$$a^2 + b^2 = c^2$$ $$6^2 + 8^2 = c^2$$ $$36 + 64 = c^2$$ $$100 = c^2$$ $$c = 10$$
Notice that 6, 8, and 10 are just double the values 3, 4, 5. Any multiple of a valid set of sides works.
Not every problem works out so neatly. Most of the time the answer is an irrational number — a square root that doesn't simplify to a whole number. That's fine; just leave it as a square root or approximate with a decimal.
Example: A right triangle has legs of 5 and 7. Find the hypotenuse.
$$5^2 + 7^2 = c^2$$ $$25 + 49 = c^2$$ $$74 = c^2$$ $$c = \sqrt{74} \approx 8.6$$
There's no cleaner form for \(\sqrt{74}\) — it just doesn't factor into a perfect square. So \(\sqrt{74}\) (or \(\approx 8.6\)) is the answer.
Finding a Missing Leg
If you know the hypotenuse and one leg, you can solve for the other leg by rearranging the formula.
Example: A right triangle has a hypotenuse of 17 and one leg of 8. Find the other leg.
$$a^2 + b^2 = c^2$$ $$8^2 + b^2 = 17^2$$ $$64 + b^2 = 289$$ $$b^2 = 225$$ $$b = 15$$
The missing leg is 15. You can verify: \(8^2 + 15^2 = 64 + 225 = 289 = 17^2\). ✓
Example: A right triangle has a hypotenuse of 11 and one leg of 5. Find the other leg.
$$5^2 + b^2 = 11^2$$ $$25 + b^2 = 121$$ $$b^2 = 96$$ $$b = \sqrt{96}$$
You can simplify \(\sqrt{96}\) because \(96 = 16 \times 6\):
$$b = \sqrt{16 \times 6} = 4\sqrt{6} \approx 9.8$$
When simplifying square roots, look for the largest perfect square that divides evenly into the number under the radical. Here \(16\) is the largest perfect square factor of \(96\), so \(\sqrt{96} = 4\sqrt{6}\).
Pythagorean Triples
A Pythagorean triple is any set of three whole numbers that satisfies \(a^2 + b^2 = c^2\). These come up frequently in problems because the sides work out without decimals or square roots. The ones worth memorizing:
| Legs | Hypotenuse |
|---|---|
| 3, 4 | 5 |
| 5, 12 | 13 |
| 8, 15 | 17 |
| 7, 24 | 25 |
Any whole-number multiple of a triple is also a triple. So 3-4-5 gives you 6-8-10, 9-12-15, 30-40-50, and so on. When a problem involves whole numbers that look familiar, check if they're a multiple of a known triple — it can save you the calculation entirely.
The Converse
The Pythagorean Theorem also works in reverse. If you have three side lengths and want to know whether they form a right triangle, plug them in and check:
- If \(a^2 + b^2 = c^2\), it's a right triangle
- If \(a^2 + b^2 > c^2\), all angles are acute (acute triangle)
- If \(a^2 + b^2 < c^2\), the largest angle is obtuse (obtuse triangle)
Example: Do sides 9, 12, and 15 form a right triangle?
$$9^2 + 12^2 = 81 + 144 = 225 = 15^2 \checkmark$$
Yes — it's a right triangle. (And you might notice these are the 3-4-5 triple scaled by 3.)
Example: Do sides 5, 7, and 9 form a right triangle?
$$5^2 + 7^2 = 25 + 49 = 74 \neq 81 = 9^2$$
Not a right triangle. Since \(74 < 81\), the angle opposite the side of length 9 is obtuse.
Quick check: what type of triangle has sides 6, 8, and 11? Show answer\(6^2 + 8^2 = 36 + 64 = 100\), and \(11^2 = 121\). Since \(100 < 121\), this is an obtuse triangle.
Where It Shows Up
Once you know this theorem, you'll notice it everywhere. The distance formula is just the Pythagorean Theorem applied to coordinate geometry — the horizontal and vertical distances between two points become the legs, and the straight-line distance is the hypotenuse. Special right triangles like the 30-60-90 get their side ratios directly from the theorem. And the Law of Cosines is essentially a generalization of the Pythagorean Theorem that applies to any triangle, not just right ones — if you need to solve a non-right triangle, the Triangle Calculator handles it step by step.
Practice
1. A right triangle has legs of 9 and 40. Find the hypotenuse. Show answer\(9^2 + 40^2 = 81 + 1600 = 1681 = 41^2\). Hypotenuse = 41. (This is the 9-40-41 triple.)
2. The hypotenuse of a right triangle is 26 and one leg is 10. Find the other leg. Show answer\(10^2 + b^2 = 26^2 \Rightarrow 100 + b^2 = 676 \Rightarrow b^2 = 576 \Rightarrow b = 24\). (The 10-24-26 triple is 5-12-13 scaled by 2.)
3. A 15-foot ladder leans against a wall. The base of the ladder is 6 feet from the wall. How high up the wall does the ladder reach? Round to one decimal place. Show answer\(6^2 + h^2 = 15^2 \Rightarrow 36 + h^2 = 225 \Rightarrow h^2 = 189 \Rightarrow h = \sqrt{189} \approx \mathbf{13.7}\) feet.