Core answer: In a right triangle, a²+b²=c² (c is the hypotenuse). Given any two sides: hypotenuse c=√(a²+b²), leg a=√(c²−b²). Example: legs 3 and 4 → hypotenuse =√(9+16)=√25=5.
The formula
Let the two legs be a and b, and the hypotenuse (the side opposite the right angle, always the longest) be c:
a² + b² = c²
Three uses:
- Find the hypotenuse: c = √(a² + b²)
- Find a leg: a = √(c² − b²)
- Test for a right angle: if the three sides satisfy the equation, the triangle is right-angled (converse theorem)
Common Pythagorean triples
Integer-sided right triangles appear constantly in exams and on job sites — and every integer multiple works too:
| Triple | ×2 | ×3 |
|---|---|---|
| 3-4-5 | 6-8-10 | 9-12-15 |
| 5-12-13 | 10-24-26 | 15-36-39 |
| 8-15-17 | 16-30-34 | — |
| 7-24-25 | 14-48-50 | — |
| 20-21-29 | — | — |
Memorize the 3-4-5 family and you can square a wall corner or check a frame without a calculator.
Worked example 1: TV size from its diagonal
A 55-inch TV (55″ diagonal) has a 16:9 screen ratio. What are its actual width and height?
- Let width = 16x, height = 9x; then (16x)² + (9x)² = 55²
- 337x² = 3025 → x = 3.0
- Width = 48″ ≈ 121.8 cm, height = 27″ ≈ 68.6 cm
Do this before buying a TV stand or squeezing the box into an elevator — far better than eyeballing it in the store.
Worked example 2: how long a ladder do you need?
To reach a 4 m eave, safety rules call for a ~75° angle (foot of the ladder about 1/4 of the height away from the wall):
- Foot distance = 4 ÷ 4 = 1 m
- Ladder² = 4² + 1² = 17
- Ladder = √17 ≈ 4.12 m, plus ~1 m extending past the eave → choose a 5 m ladder
The converse: testing for a right angle
Three planks of 6, 8 and 10 m are nailed into a triangle. Is it right-angled?
6² + 8² = 36 + 64 = 100 = 10² ✓ Yes — the right angle sits between the 6 m and 8 m sides.
This is exactly how contractors square a foundation: measure 3 m along one wall, 4 m along the other; if the diagonal reads exactly 5 m, the corner is 90°.
Real-world applications
| Scenario | How to use it |
|---|---|
| Squaring a wall corner | 3-4-5 string method to verify 90° |
| Straight-line distance | east-west gap a, north-south gap b → √(a²+b²) |
| TV/screen sizing | inches = diagonal; combine with aspect ratio for width/height |
| Ladder/ramp length | height + horizontal distance → slope length |
| Fitting furniture in an elevator | item diagonal ≤ elevator diagonal |
Common mistakes and myths
- "It works for any triangle" — right triangles only. For others use the law of cosines: c²=a²+b²−2ab·cosC.
- Picking the wrong side as hypotenuse — the hypotenuse is always the longest side, opposite the right angle; c must sit alone on one side of the equation.
- Adding instead of subtracting for a leg — a=√(c²−b²) uses a minus sign; the result must be positive to be valid.
- Mixing units — unify meters and centimeters before squaring; the unit of area gets squared too.
Use the [Pythagorean Theorem Calculator](/c/geometry/pythagorean) to get the third side from any two, and the [Triangle Calculator](/c/geometry/triangle-solver) for non-right triangles.