Hypotenuse Calculator
Enter the two legs of a right triangle to find the hypotenuse instantly. This tool applies the Pythagorean theorem and also gives you the triangle's area.
The Pythagorean Theorem and the Hypotenuse
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². This relationship has been known for millennia and forms the foundation of right triangle geometry.
To find the hypotenuse, square each leg, add the results, and take the square root. For example, with legs 3 and 4, you compute 9 + 16 = 25, then √25 = 5. The calculation works for any right triangle, regardless of size or orientation.
This calculator automates the arithmetic and rounds the result to four decimal places for precision. It also computes the triangle's area, which equals half the product of the two legs.
Real-World Hypotenuse Applications
Construction workers use the hypotenuse to check if corners are square and to calculate diagonal bracing lengths. The classic 3-4-5 triangle provides a quick field check: measure 3 feet along one edge, 4 feet along the perpendicular edge, and verify the diagonal is 5 feet.
In screen manufacturing, the diagonal measurement (hypotenuse) is the standard way to specify size. A 50-inch TV has a 50-inch diagonal, and you can compute the width and height from the aspect ratio using the Pythagorean theorem.
Navigation systems calculate straight-line distances by treating north-south and east-west movements as legs of a right triangle. The hypotenuse gives the shortest path between two points, which is essential for route optimization and fuel efficiency.
Pythagorean Triples and the Hypotenuse
Pythagorean triples are sets of three whole numbers that satisfy a² + b² = c². The most famous is 3-4-5. Any multiple also works: 6-8-10, 9-12-15, 30-40-50. These triples make mental math easy and are widely used in construction and design.
Other common triples include 5-12-13, 8-15-17, and 7-24-25. Engineers and carpenters memorize several of these to speed up calculations and verify measurements without a calculator.
Most right triangles do not have whole-number sides. For example, a triangle with legs 1 and 1 has hypotenuse √2 (about 1.4142), an irrational number. This calculator handles both integer and non-integer cases with equal precision.
Frequently Asked Questions
What is the hypotenuse?
The hypotenuse is the longest side of a right triangle, opposite the 90-degree angle. It connects the endpoints of the two legs.
How do you calculate the hypotenuse?
Use the Pythagorean theorem: c = √(a² + b²), where a and b are the legs and c is the hypotenuse. Square each leg, add them, then take the square root.
Does the hypotenuse formula work for non-right triangles?
No. The Pythagorean theorem only applies to right triangles. For other triangles, use the law of cosines.
What are some common hypotenuse examples?
A 3-4-5 triangle has legs 3 and 4, hypotenuse 5. A 5-12-13 triangle has legs 5 and 12, hypotenuse 13. These are Pythagorean triples.
Can I find a leg if I know the hypotenuse and the other leg?
Yes. Rearrange the formula: a = √(c² - b²). This calculator focuses on finding the hypotenuse, but you can use a Pythagorean theorem calculator for missing legs.