The Formula
In a right triangle, the two sides forming the right angle are called the legs (a and b). The side across from that right angle, always the longest of the three, is the hypotenuse (c).
Legs 3 and 4 → hypotenuse:
3² + 4² = c²
9 + 16 = c²
c = √25 = 5
Finding a Missing Leg
Rearrange the same formula when it's a leg you're solving for instead of the hypotenuse:
Hypotenuse = 13, one leg = 5:
a = √(169 − 25) = √144 = 12
Common Triples
A Pythagorean triple is a set of three whole numbers that fits a² + b² = c² exactly. A handful of them come up again and again in textbook problems:
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 6 | 8 | 10 |
| 9 | 40 | 41 |
Multiply every number in a triple by the same factor and it still works: 6-8-10 is the 3-4-5 triple doubled, and 15-20-25 is that same triple scaled by 5.
Is This a Right Triangle?
You can test any triangle for a right angle just by plugging its three side lengths into the formula. Square the two shorter sides, add them, then compare the total to the square of the longest side - if they match, it's a right triangle:
6² + 8² = 36 + 64 = 100 = 10² ✓ Right triangle
Sides 4, 5, 7:
4² + 5² = 16 + 25 = 41 ≠ 49 (7²) ✗ Not a right triangle
Where This Shows Up in Real Life
This isn't a formula that stays confined to the classroom - it solves practical problems too:
| Application | How it's used |
|---|---|
| Construction | Checking if corners are perfectly square (3-4-5 method) |
| Navigation | Finding shortest distance between two points |
| Screen size | TV and monitor sizes are measured diagonally |
| Staircase design | Calculating stringer length from rise and run |
| Physics | Combining perpendicular velocity components |
The Distance Formula Connection
The distance formula you'll meet in coordinate geometry is really this same theorem wearing a different name. Treat the horizontal gap between two points as one leg and the vertical gap as the other, and the straight-line distance between them works out to be the hypotenuse:
From (1, 2) to (4, 6):
Horizontal gap = 3, vertical gap = 4
Distance = √(9 + 16) = √25 = 5
Solving It With a Calculator
SolverCalc's triangle calculator applies this theorem automatically, switching to the law of cosines whenever the triangle isn't a right triangle, to find missing sides and angles. Type in what you already know and it fills in the rest, alongside a live diagram of the result.
Key Takeaways
a² + b² = c² works for every right triangle, full stop. Reach for it to solve for a missing side, test whether a triangle actually has a right angle, or work out the distance between two points on a grid. Learn the common triples, 3-4-5, 5-12-13, 8-15-17, and you'll recognize them the moment they show up in a problem.