Hypotenuse Calculator
Calculate the Hypotenuse
Enter the lengths of the two shorter sides (a and b) of a right-angled triangle to find the length of the hypotenuse (c) using our Hypotenuse Calculator.
Results:
Side a²: 9.00
Side b²: 16.00
a² + b²: 25.00
Understanding the Hypotenuse Calculator
What is a Hypotenuse Calculator?
A Hypotenuse Calculator is a tool used to find the length of the hypotenuse of a right-angled triangle. The hypotenuse is the longest side of a right-angled triangle, and it is the side opposite the right angle (90-degree angle). This calculator utilizes the Pythagorean theorem (a² + b² = c²) to determine the length of the hypotenuse (c) when the lengths of the other two sides (a and b, often called legs) are known.
Anyone working with right-angled triangles can benefit from a Hypotenuse Calculator. This includes students learning geometry or trigonometry, engineers, architects, carpenters, builders, and even DIY enthusiasts planning projects that involve right angles. It simplifies the calculation and reduces the chance of manual errors.
A common misconception is that the hypotenuse is just any long side of any triangle. However, the term "hypotenuse" specifically applies *only* to the longest side of a *right-angled* triangle. Our Hypotenuse Calculator is designed for these specific triangles.
Hypotenuse Calculator Formula and Mathematical Explanation
The Hypotenuse Calculator is based on the Pythagorean theorem, a fundamental principle in Euclidean geometry. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
The formula is:
c² = a² + b²
To find the length of the hypotenuse (c), we take the square root of both sides:
c = √(a² + b²)
Here's a step-by-step derivation:
- Square the length of side a: Calculate a².
- Square the length of side b: Calculate b².
- Sum the squares: Add a² and b² together (a² + b²).
- Take the square root: Find the square root of the sum (√(a² + b²)), which gives you the length of the hypotenuse c.
Our Hypotenuse Calculator performs these steps automatically.
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Length of one leg (shorter side) | Any unit of length (e.g., cm, m, inches, feet) | Positive numbers |
| b | Length of the other leg (shorter side) | Same unit as 'a' | Positive numbers |
| c | Length of the hypotenuse (longest side) | Same unit as 'a' and 'b' | Positive numbers, c > a and c > b |
Practical Examples (Real-World Use Cases)
The Hypotenuse Calculator is useful in various real-world scenarios:
Example 1: Screen Size
You want to know the diagonal screen size of a monitor that is 16 inches wide (a) and 9 inches high (b). The diagonal is the hypotenuse.
- Side a = 16 inches
- Side b = 9 inches
Using the Hypotenuse Calculator (or formula c = √(16² + 9²)):
c = √(256 + 81) = √337 ≈ 18.36 inches. So, the screen is approximately an 18.4-inch screen diagonally.
Example 2: Ladder Against a Wall
You need to place a ladder against a wall to reach a height of 8 feet (a). You place the base of the ladder 6 feet away from the wall (b). What is the length of the ladder (hypotenuse c)?
- Side a = 8 feet
- Side b = 6 feet
Using the Hypotenuse Calculator (or formula c = √(8² + 6²)):
c = √(64 + 36) = √100 = 10 feet. The ladder needs to be 10 feet long.
Find more tools like our Online Geometry Tools for other shapes.
How to Use This Hypotenuse Calculator
Using our Hypotenuse Calculator is straightforward:
- Enter Side a: Input the length of one of the shorter sides of the right-angled triangle into the "Length of Side a" field.
- Enter Side b: Input the length of the other shorter side into the "Length of Side b" field. Ensure you use the same units for both sides.
- View Results: The calculator will instantly display the length of the hypotenuse (c), as well as the intermediate values of a², b², and a²+b². The formula c = √(a² + b²) is also shown.
- Visual Aid: The diagram below the results visually represents a triangle with the entered side lengths (labels update, though the visual triangle is scaled to fit).
- Reset: Click the "Reset" button to clear the inputs and results to their default values.
- Copy: Click "Copy Results" to copy the main result and intermediate values to your clipboard.
When reading the results, the "Hypotenuse (c)" is the primary value you are looking for. The intermediate values help you understand the steps of the calculation based on the Pythagorean Theorem Explained.
Key Factors That Affect Hypotenuse Calculation Results
The accuracy and relevance of the hypotenuse calculation depend on several factors:
- Accuracy of Input Measurements: The most critical factor. If the lengths of sides 'a' and 'b' are measured inaccurately, the calculated hypotenuse will also be inaccurate. Use precise measuring tools.
- Assuming a Perfect Right Angle: The Pythagorean theorem and this Hypotenuse Calculator only apply to triangles with a perfect 90-degree angle between sides 'a' and 'b'. If the angle is not exactly 90 degrees, the formula is not directly applicable.
- Units of Measurement: Ensure that the lengths of side 'a' and side 'b' are entered using the same units (e.g., both in cm, or both in inches). The hypotenuse will be in the same unit.
- Rounding: The calculator may round the result to a certain number of decimal places. For very high precision requirements, be aware of the level of rounding. Our calculator typically shows two decimal places.
- Real-World vs. Ideal Conditions: In practical applications like construction, materials have thickness, and perfect right angles might be hard to achieve, introducing slight deviations from the calculated value.
- Scale of the Problem: For very large distances (e.g., astronomical) or very small scales (e.g., subatomic), other geometric principles or adjustments might be needed, though for everyday use, the Pythagorean theorem is highly accurate. For more on triangle properties, see our Right Triangle Properties guide.
Frequently Asked Questions (FAQ)
The hypotenuse is the longest side of a right-angled triangle, located opposite the right angle.
It's a formula (a² + b² = c²) that relates the lengths of the sides of a right-angled triangle, where 'a' and 'b' are the shorter sides (legs), and 'c' is the hypotenuse.
No, this calculator is specifically for right-angled triangles. For other triangles, you would need different formulas (like the Law of Sines or Cosines).
You can use any unit of length (cm, meters, inches, feet, etc.), as long as you use the same unit for both side 'a' and side 'b'. The result for the hypotenuse will be in that same unit.
The calculator will show an error or produce an invalid result (like NaN) because side lengths cannot be negative.
The calculator performs the mathematical operation accurately. The accuracy of the result depends entirely on the accuracy of your input measurements for sides 'a' and 'b'.
Yes, if you know the hypotenuse and one other side, you can rearrange the formula to find the missing side (e.g., a = √(c² – b²)). You might need a Side Length Calculator for that.
It's used in navigation (calculating shortest distance), construction (ensuring square corners), architecture, surveying, and even in fields like computer graphics and physics.