📐 Hypotenuse Calculator
Calculate the hypotenuse of a right triangle using the Pythagorean theorem
How to Use This Calculator
Enter Two Leg Lengths
Input the lengths of the two legs (the perpendicular sides) of the right triangle.
Calculate
Click "Calculate Hypotenuse" to find the length of the hypotenuse using the Pythagorean theorem.
Review Results
See the hypotenuse length, along with area, perimeter, and angles of the right triangle.
Pythagorean Theorem
c² = a² + b²
Hypotenuse: c = √(a² + b²)
Pythagorean Theorem:
In a right triangle, the square of the hypotenuse equals the sum of squares of the two legs.
c² = a² + b²
To Find Hypotenuse:
c = √(a² + b²)
Take the square root of the sum of squares of the legs.
Example: If leg 1 = 3 and leg 2 = 4:
c = √(3² + 4²) = √(9 + 16) = √25 = 5
Special Cases:
- If legs are equal (a = b), then c = a√2 (45-45-90 triangle)
- If one leg is half the hypotenuse, it's a 30-60-90 triangle
- 3-4-5, 5-12-13, 8-15-17 are common Pythagorean triples
About Hypotenuse Calculator
The Hypotenuse Calculator finds the length of the hypotenuse (longest side) of a right triangle using the Pythagorean theorem. The hypotenuse is the side opposite the right angle and is always the longest side.
When to Use This Calculator
- Geometry: Find the hypotenuse in right triangle problems
- Construction: Calculate diagonal distances or lengths
- Education: Learn and practice the Pythagorean theorem
- Engineering: Calculate dimensions in right triangular structures
- Trigonometry: Find hypotenuse before calculating trigonometric ratios
- Real Estate: Calculate diagonal measurements of rectangular areas
Why Use Our Calculator?
- ✅ Quick Calculation: Instantly find hypotenuse from two legs
- ✅ Step-by-Step Display: See the calculation process
- ✅ Complete Information: Shows area, perimeter, and angles
- ✅ Accurate: Uses precise Pythagorean theorem
- ✅ Educational: Helps understand right triangles
- ✅ Free: No registration required
Key Concepts
- Right Triangle: A triangle with one 90° angle
- Legs: The two sides that form the right angle
- Hypotenuse: The side opposite the right angle, always the longest
- Pythagorean Theorem: c² = a² + b², where c is hypotenuse
- Always Longest: Hypotenuse is always longer than either leg
Example
If leg 1 = 6 units and leg 2 = 8 units:
- Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 units
- Area = (6 × 8) / 2 = 24 square units
- Perimeter = 6 + 8 + 10 = 24 units
Frequently Asked Questions
What is the hypotenuse?
The hypotenuse is the longest side of a right triangle, located opposite the right angle. It's always longer than either of the two legs.
How do you calculate the hypotenuse?
Use the Pythagorean theorem: c = √(a² + b²), where a and b are the lengths of the two legs, and c is the hypotenuse.
Can the hypotenuse be shorter than a leg?
No! The hypotenuse is always the longest side of a right triangle. This is guaranteed by the Pythagorean theorem: c² = a² + b² means c > a and c > b.
What if I know the hypotenuse and one leg?
You can find the other leg using: b = √(c² - a²). Rearrange the Pythagorean theorem: a² + b² = c², so b² = c² - a².
Does this work for non-right triangles?
No, the Pythagorean theorem only applies to right triangles. For other triangles, use the Law of Cosines.
What are Pythagorean triples?
Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a² + b² = c². Common examples: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25).
Why is it called the hypotenuse?
The word "hypotenuse" comes from the Greek "hypoteinousa" meaning "stretching under" (the right angle). It's the side that "stretches under" or subtends the right angle.