Hypotenuse Calculator
Easily find the hypotenuse of a right-angled triangle with our online hypotenuse calculator. Enter the lengths of the two shorter sides (a and b) below.
Calculate Hypotenuse (c)
Enter the length of the first shorter side.
Enter the length of the second shorter side.
Visual representation of sides a, b, and hypotenuse c.
What is a Hypotenuse?
The hypotenuse is the longest side of a right-angled triangle. It is the side opposite the right angle (the 90-degree angle). The other two sides are called the legs (or cathetus, plural catheti). The relationship between the lengths of the legs (a and b) and the hypotenuse (c) is defined by the Pythagorean theorem. Understanding the hypotenuse is fundamental in geometry and trigonometry, and our hypotenuse calculator makes finding its length straightforward.
Anyone studying geometry, trigonometry, architecture, engineering, or even doing DIY projects that involve right angles might need to calculate the hypotenuse. Our hypotenuse calculator is designed for students, teachers, professionals, and hobbyists alike.
A common misconception is that any longest side of a triangle is the hypotenuse. This is only true for right-angled triangles; the hypotenuse specifically refers to the side opposite the right angle.
Hypotenuse Calculator Formula and Mathematical Explanation
The hypotenuse calculator uses the Pythagorean theorem to find the length of the hypotenuse (c) given the lengths of the other two sides (a and b) of a right-angled triangle. The formula is:
c² = a² + b²
To find ‘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 (a²).
- Square the length of side b (b²).
- Add the results from step 1 and step 2 (a² + b²).
- Take the square root of the sum from step 3 (√(a² + b²)) to find the length of the hypotenuse c.
Our hypotenuse calculator performs these steps automatically.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Length of the first leg | Any unit of length (e.g., cm, m, inches, feet) | > 0 |
| b | Length of the second leg | Same unit as ‘a’ | > 0 |
| c | Length of the hypotenuse | Same unit as ‘a’ and ‘b’ | > a and > b |
Table showing the variables used in the hypotenuse calculation.
Practical Examples (Real-World Use Cases)
Example 1: Building a Ramp
Imagine you are building a ramp that needs to cover a horizontal distance of 12 feet (side a) and reach a height of 5 feet (side b). To find the length of the ramp surface (the hypotenuse c), you would use the hypotenuse calculator or the formula:
a = 12 feet, b = 5 feet
c = √(12² + 5²) = √(144 + 25) = √169 = 13 feet
The ramp surface will be 13 feet long.
Example 2: Screen Size
A TV screen is advertised as 50 inches, which is the diagonal length (hypotenuse). If the screen’s width (side a) is 43.6 inches, what is its height (side b)? We can rearrange the formula: b = √(c² – a²).
c = 50 inches, a = 43.6 inches
b = √(50² – 43.6²) = √(2500 – 1900.96) = √599.04 ≈ 24.48 inches
While our tool is primarily a hypotenuse calculator (finding c), the principle is the same. You could use the inputs to find a missing side if you know the hypotenuse and one side by working backward or using a more comprehensive triangle solver.
How to Use This Hypotenuse Calculator
- 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 automatically display the length of the hypotenuse (c), as well as the squares of side a and side b, and their sum. The results update in real-time as you type. The chart will also update.
- Reset: Click the “Reset” button to clear the inputs and results and return to the default values.
- Copy Results: Click “Copy Results” to copy the lengths of a, b, c, and the intermediate squares to your clipboard.
When reading the results from the hypotenuse calculator, remember that the hypotenuse ‘c’ will always be longer than both ‘a’ and ‘b’. The units of ‘c’ will be the same as the units you used for ‘a’ and ‘b’.
Key Factors That Affect Hypotenuse Calculation Results
The calculation of the hypotenuse is directly and solely dependent on the lengths of the other two sides (the legs) of the right-angled triangle. Here are the key factors:
- Length of Side a: The longer side ‘a’ is, the longer the hypotenuse ‘c’ will be, assuming ‘b’ remains constant.
- Length of Side b: Similarly, the longer side ‘b’ is, the longer the hypotenuse ‘c’ will be, assuming ‘a’ remains constant.
- The Right Angle: The formula c² = a² + b² is only valid because the triangle is a right-angled triangle. If the angle between sides a and b is not 90 degrees, this formula and our hypotenuse calculator based on it do not apply. For other triangles, you’d need the Law of Cosines. See our geometry formulas page for more.
- Units of Measurement: Consistency is key. If you measure side ‘a’ in centimeters and side ‘b’ in inches, the result from the hypotenuse calculator will be meaningless unless you first convert them to the same unit.
- Accuracy of Input Values: The accuracy of the calculated hypotenuse depends directly on the accuracy of the measurements of sides ‘a’ and ‘b’. Small errors in input can lead to small errors in the output.
- The Pythagorean Theorem: The entire calculation is based on this fundamental theorem. Our hypotenuse calculator is a direct application of it.
Frequently Asked Questions (FAQ)
- What is the Pythagorean theorem?
- The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs). The formula is a² + b² = c².
- Can I use the hypotenuse calculator for any triangle?
- No, this hypotenuse calculator is specifically for right-angled triangles because it uses the Pythagorean theorem. For non-right-angled triangles, you would need different formulas like the Law of Sines or the Law of Cosines.
- What units can I use with the hypotenuse calculator?
- You can use any unit of length (cm, meters, inches, feet, etc.), but you MUST use the same unit for both side a and side b. The hypotenuse will be calculated in the same unit.
- How do I find the length of a leg if I know the hypotenuse and the other leg?
- You can rearrange the Pythagorean theorem: a = √(c² – b²) or b = √(c² – a²). While this tool is primarily a hypotenuse calculator (finds c), you can use the formula to find a or b manually.
- Does the order of side a and side b matter?
- No, because addition is commutative (a² + b² = b² + a²), it doesn’t matter which leg you label as ‘a’ and which you label as ‘b’ in the hypotenuse calculator.
- What if I enter zero or negative values?
- The lengths of the sides of a triangle must be positive numbers. The hypotenuse calculator will show an error or not calculate if you enter zero or negative values for the sides.
- Is the hypotenuse always the longest side?
- Yes, in a right-angled triangle, the hypotenuse is always the longest side.
- Where is the Pythagorean theorem used in real life?
- It’s used extensively in construction, architecture, navigation, surveying, engineering, and even computer graphics to calculate distances and angles. Our Pythagorean theorem explanation page has more details.
Related Tools and Internal Resources
- Pythagorean Theorem Explained: A detailed look at the theorem behind the hypotenuse calculator.
- Right-Angled Triangle Area Calculator: Calculate the area of a right triangle given its sides.
- Triangle Solver: A more comprehensive tool for solving various triangle problems, not just finding the hypotenuse.
- Geometry Formulas: A collection of useful formulas for various geometric shapes.
- Math Calculators Hub: Explore our full range of online math tools and math calculators.
- Online Calculators: Our main page featuring various calculators.