Loading the page...
Preparing tools and content for you. This usually takes a second.
Preparing tools and content for you. This usually takes a second.
Fetching calculator categories and tools for this section.
Find the longest side of a right triangle instantly using the Pythagorean theorem (a² + b² = c²) with detailed step-by-step solutions.
Master the core concepts of geometry and the Pythagorean theorem.
The Pythagorean equation is written as a² + b² = c². To solve for c (the hypotenuse), you take the square root of the sum of a² and b², making it c = √(a² + b²).
The hypotenuse is strictly the longest side of a right triangle. If you calculate a hypotenuse that is shorter than either of your legs, you\'ve made an error in calculation.
This formula is used daily in architecture, construction, navigation, and surveying to find the shortest straight-line distance between two points.
The hypotenuse is the side opposite the right angle in a right triangle, and it is always the longest side. Calculating it quickly helps in geometry, construction layout, surveying, navigation, and many engineering workflows.
This metric matters because diagonal distance is often the value you cannot measure directly in the field. With two known perpendicular legs, the hypotenuse gives you the exact straight-line span needed for planning and precision work.
a² + b² = c²
Here, c is the hypotenuse and a, b are the two legs.
c = sqrt(a² + b²)
Square each leg, add results, then take the positive square root.
Legs: 3 and 4
Hypotenuse: 5
Common in field checks for right angles.
Wall height: 12 ft
Base offset: 5 ft
Ladder length: 13 ft
Ensures proper ladder selection and placement.
Room: 10 ft by 12 ft
Hypotenuse: ~15.62 ft
Useful for furniture fit or cable runs across corners.
Compare common leg pairs and resulting hypotenuse values to speed up geometry work and real-world estimations.
| Leg A | Leg B | Hypotenuse (c) | Use Case |
|---|---|---|---|
| 3 | 4 | 5 | Quick right-angle verification |
| 5 | 12 | 13 | Framing and ladder planning |
| 8 | 15 | 17 | Layout and surveying tasks |
| 10 | 24 | 26 | Large-span geometric checks |
| Leg A | Leg B | Hypotenuse (c) | Use Case | | --- | --- | --- | --- | | 3 | 4 | 5 | Quick right-angle verification | | 5 | 12 | 13 | Framing and ladder planning | | 8 | 15 | 17 | Layout and surveying tasks | | 10 | 24 | 26 | Large-span geometric checks |
The hypotenuse is the longest side of a right-angled triangle. It is always the side that is opposite the right angle (the 90-degree angle).
The Pythagorean theorem is a fundamental relation in geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides: a² + b² = c².
No, the Pythagorean theorem (and this calculator) only works for right-angled triangles—triangles where one angle is exactly 90 degrees. For other triangles, you would need to use the Law of Cosines.
If you know the hypotenuse (c) and one leg (let's say a), you can rearrange the formula to find the other leg (b): b = √(c² - a²). Our current calculator is specifically designed to find the hypotenuse when both legs are known.
In physical geometry, side lengths are strictly positive distances. Though algebra yields both positive and negative roots for c² = a² + b², we always take the principal (positive) square root for the hypotenuse.
A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². The most famous example is 3, 4, 5. Checking for triples is a quick way to find side lengths without calculation.
No. According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the remaining side. Thus, the hypotenuse is always less than a + b.
If the two legs are equal (a = b), the triangle is an isosceles right triangle (a 45-45-90 triangle). The hypotenuse formula simplifies to c = a√2. You simply multiply the leg length by the square root of 2.
It depends on your context. In a math class, your teacher might want the exact radical form (e.g., √20 = 2√5). In engineering or woodworking, a rounded decimal approximation (e.g., 4.47) is usually preferred.
The units can be anything (inches, centimeters, meters, miles), as long as both legs share the exact same unit. The final hypotenuse will automatically be in that same unit.
Explore more tools to help you with shapes, angles, and distances.
Calculate total, lateral, and base surface area of cylinders.
Simplify square roots to radical form with perfect square factorization.
Calculate roots of polynomials with step-by-step solutions.
Calculate tangent values for angles in degrees or radians.
Convert between radians and degrees with π expressions.
Find all factors, factor pairs, and the prime factorization of any positive integer.
Suggested hashtags: #hypotenuse #geometry #math #pythagoreantheorem #calculator