Hypotenuse for a Right Triangle with legs 9 and 4
The hypotenuse, area, and perimeter of a right triangle with legs 9 and 4, using the Pythagorean theorem. Adjust any field below to try your own numbers.
Leg A
9
Leg B
4
Hypotenuse
9.8489
With legs of 9 and 4, the Pythagorean theorem (a² + b² = c²) gives a hypotenuse of 9.8489. The resulting right triangle has an area of 18 and a perimeter of 22.8489.
Area
18
Perimeter
22.8489
What is the Pythagorean Theorem Calculator?
The Pythagorean theorem calculator finds the missing side of a right triangle when you know the other two. It's one of the oldest and most widely used relationships in geometry, connecting the two shorter sides (legs) of a right triangle to its longest side (the hypotenuse) through a single, elegant equation.
Beyond geometry homework, the same relationship underlies distance calculations in construction (checking that a corner is truly square), navigation, computer graphics, and physics — anywhere two perpendicular measurements need to be combined into a single straight-line distance.
Common Pythagorean Triples
A Pythagorean triple is a set of three positive integers a, b, c that satisfy a² + b² = c² exactly, with no rounding. Your entered legs are highlighted below if they match a well-known triple.
| Leg A | Leg B | Hypotenuse C |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 20 | 21 | 29 |
| 9 | 40 | 41 |
| 12 | 35 | 37 |
| 11 | 60 | 61 |
| 6 | 8 | 10 |
| 9 | 12 | 15 |
| 10 | 24 | 26 |
| 15 | 20 | 25 |
The Pythagorean Theorem
Named after the ancient Greek mathematician Pythagoras (though the relationship was known and used by Babylonian and Indian mathematicians centuries earlier), the theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs.
Solving for any single unknown just requires rearranging: c = √(a² + b²) when solving for the hypotenuse, or a = √(c² − b²) when solving for a leg. The theorem only applies to right triangles — triangles containing a 90° angle — not to triangles in general.
Why Pythagorean Triples Matter
Most combinations of two whole-number legs produce an irrational hypotenuse — for example, legs of 2 and 3 give a hypotenuse of √13 ≈ 3.606, which never terminates or repeats. A Pythagorean triple is the rare exception: a set of three whole numbers that satisfy the theorem exactly, with no rounding at all. The (3, 4, 5) triple is by far the most famous and is the smallest possible triple — it's been used since antiquity by builders and surveyors to check that a corner is exactly square, since any multiple of 3-4-5 (like 6-8-10 or 9-12-15) also works.
Real-World Uses of the Pythagorean Theorem
Carpenters and construction crews use the "3-4-5 rule" to square corners on job sites without a protractor: measure 3 feet along one wall and 4 feet along the perpendicular wall, and if the diagonal between those two points measures exactly 5 feet, the corner is a true right angle. The same principle scales to any multiple (30-40-50 feet, for example) for larger structures.
In everyday navigation and mapping, the theorem converts two perpendicular distances — say, blocks traveled north and blocks traveled east — into a single straight-line ("as the crow flies") distance. TV and monitor sizes are also measured diagonally using the same relationship: a 16:9 aspect-ratio screen's width, height, and diagonal size form a right triangle.
Example — Your Current Inputs
With legs of 9 and 4, the Pythagorean theorem (a² + b² = c²) gives a hypotenuse of 9.8489. The resulting right triangle has an area of 18 and a perimeter of 22.8489.
Additional Example — A TV Screen
A television is advertised as "55 inches" — that number is always the diagonal measurement. For a 16:9 widescreen TV, the width works out to about 47.9 inches and the height to about 27.0 inches. Checking with the Pythagorean theorem: 47.9² + 27.0² = 2294.4 + 729.0 = 3023.4, and √3023.4 ≈ 55.0 — confirming the advertised diagonal size.
About These Parameters
- Leg A and Leg B
- The two sides that meet at the right angle. Enter both of these to solve for the hypotenuse, or enter one leg plus the hypotenuse to solve for the other leg.
- Hypotenuse (C)
- The longest side, always opposite the right angle. Leave this field blank if you want the calculator to solve for it from the two legs.
Frequently Asked Questions
Does the Pythagorean theorem work on any triangle?
No — it only applies to right triangles, which contain exactly one 90° angle. For triangles without a right angle, you need the Law of Cosines instead, which generalizes the same idea with an extra term that accounts for the angle between the two known sides. Try the Triangle Calculator for general (non-right) triangles.
How do I know which side is the hypotenuse?
The hypotenuse is always the longest side of a right triangle, and it's always the side directly opposite (across from) the 90° angle — never one of the two sides that form the right angle itself.
Can I use the theorem to check if a triangle is a right triangle?
Yes — given all three sides, compute a² + b² for the two shorter sides and compare it to c² for the longest side. If they're equal, the triangle is a right triangle; if a² + b² is greater than c², the triangle is acute; if it's less, the triangle is obtuse. This is called the converse of the Pythagorean theorem.
Why are there infinitely many Pythagorean triples?
Any multiple of an existing triple is also a valid triple — doubling (3, 4, 5) gives (6, 8, 10), which still satisfies 6² + 8² = 10². Beyond simple multiples, there are also infinitely many "primitive" triples (where the three numbers share no common factor), generated by formulas involving two arbitrary coprime integers of opposite parity — a result proven by Euclid over 2,000 years ago.