Hypotenuse of a Triangle Calculator
Easily find the length of the hypotenuse (the longest side) of a right-angled triangle using our simple hypotenuse of a triangle calculator. Enter the lengths of the other two sides (A and B) below.
Calculate Hypotenuse
Hypotenuse (Side C):
5 unitsSide A Squared (a²): 9
Side B Squared (b²): 16
Sum of Squares (a² + b²): 25
Comparison of side lengths A, B, and the Hypotenuse (C).
What is the Hypotenuse of a Triangle?
The hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle (the 90-degree angle). It connects the two legs (the shorter sides) of the right triangle. The term "hypotenuse" comes from the Greek word 'hypoteinousa,' meaning 'stretching under' (the right angle). Understanding the hypotenuse is fundamental in geometry and trigonometry, particularly when using the Pythagorean theorem. A hypotenuse of a triangle calculator simplifies finding this length.
Anyone studying geometry, trigonometry, architecture, engineering, or even DIY enthusiasts might need to calculate the hypotenuse. For example, determining the length of a diagonal brace or the shortest distance between two points offset horizontally and vertically involves finding the hypotenuse. This hypotenuse of a triangle calculator is a handy tool for these tasks.
A common misconception is that any longest side of any triangle is a hypotenuse. This is incorrect; the term hypotenuse applies *only* to right-angled triangles.
Hypotenuse of a Triangle Formula and Mathematical Explanation
The length of the hypotenuse (c) of a right-angled triangle is found using the Pythagorean theorem, named after the ancient Greek mathematician Pythagoras. The theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides (legs), a and b.
The formula is:
c² = a² + b²
To find the hypotenuse c, we take the square root of both sides:
c = √(a² + b²)
Where:
- c = length of the hypotenuse
- a = length of one leg (side A)
- b = length of the other leg (side B)
The hypotenuse of a triangle calculator automates this calculation.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Length of Side A (one leg) | Any unit of length (e.g., cm, m, inches, feet) | Positive numbers |
| b | Length of Side B (the other leg) | Same unit as 'a' | Positive numbers |
| c | Length of the Hypotenuse | Same unit as 'a' and 'b' | Greater than 'a' and 'b' |
Variables used in the Pythagorean theorem.
Practical Examples (Real-World Use Cases)
Example 1: Finding the Diagonal of a Screen
You have a rectangular TV screen that is 48 inches wide (side a) and 27 inches high (side b). You want to find the diagonal length of the screen, which is the hypotenuse of the right triangle formed by the width, height, and diagonal.
- Side A (a) = 48 inches
- Side B (b) = 27 inches
Using the formula c = √(48² + 27²):
c = √(2304 + 729) = √3033 ≈ 55.07 inches
The diagonal length of the screen is approximately 55.07 inches. Our hypotenuse of a triangle calculator can give you this instantly.
Example 2: Calculating Roof Rafter Length
An architect is designing a roof with a rise (height, side a) of 9 feet and a run (horizontal distance, side b) of 12 feet. They need to find the length of the rafter (the hypotenuse).
- Side A (a) = 9 feet
- Side B (b) = 12 feet
Using the formula c = √(9² + 12²):
c = √(81 + 144) = √225 = 15 feet
The length of the rafter needs to be 15 feet. A reliable Pythagorean theorem calculator is essential here.
How to Use This Hypotenuse of a Triangle Calculator
Using our hypotenuse of a triangle 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 as Side A.
- Specify Units (Optional): Enter the unit of measurement (e.g., cm, meters, inches, feet) in the "Units" field. This is for labeling the result and does not affect the numerical calculation, as long as Side A and Side B use the same units.
- Calculate: The calculator automatically updates the hypotenuse value and intermediate calculations as you type. You can also click the "Calculate" button.
- View Results: The "Hypotenuse (Side C)" will be displayed prominently, along with the intermediate values (a², b², a²+b²).
- Reset: Click "Reset" to clear the fields and return to default values.
- Copy: Click "Copy Results" to copy the main result, intermediates, and unit to your clipboard.
The results from the hypotenuse of a triangle calculator give you the length of the longest side. This is crucial for construction, navigation, and various geometric problems. Also, check out our triangle angle calculator for more triangle-related calculations.
Key Factors That Affect Hypotenuse Results
The primary factors affecting the calculated hypotenuse are:
- Accuracy of Side A Measurement: Any error in measuring Side A will directly impact the hypotenuse calculation. Precise measurement is key.
- Accuracy of Side B Measurement: Similarly, the precision of the Side B measurement is crucial for an accurate hypotenuse value.
- Right Angle Assumption: The Pythagorean theorem and this hypotenuse of a triangle calculator only apply if the triangle is truly right-angled (one angle is exactly 90 degrees). If the angle is not 90 degrees, the formula is incorrect.
- Consistent Units: Sides A and B must be measured in the same units. If one is in cm and the other in meters, you must convert them to the same unit before using the calculator.
- Rounding: The final hypotenuse value might be a non-terminating decimal (like √2). The level of rounding used will affect the final displayed value. Our calculator provides a reasonably precise result.
- Calculator Precision: The internal precision of the square root function used by the calculator software can introduce very minor differences in the last decimal places.
Frequently Asked Questions (FAQ)
- What is a hypotenuse?
- The hypotenuse is the longest side of a right-angled triangle, opposite the right angle.
- What formula is used by the hypotenuse of a triangle calculator?
- It uses the Pythagorean theorem: c = √(a² + b²), where c is the hypotenuse, and a and b are the other two sides.
- Can I use this calculator for any triangle?
- No, this hypotenuse of a triangle calculator is specifically for right-angled triangles.
- Do the units of Side A and Side B matter?
- Yes, they must be the same unit (e.g., both in cm or both in inches). The calculator assumes they are the same and applies the unit you enter to the result.
- Is the hypotenuse always the longest side?
- Yes, in a right-angled triangle, the hypotenuse is always longer than either of the other two sides.
- What if I know the hypotenuse and one side, but not the other side?
- You can rearrange the formula: a = √(c² – b²) or b = √(c² – a²). You would need a right triangle calculator that allows these inputs, or rearrange it manually.
- How accurate is this hypotenuse of a triangle calculator?
- The calculator performs the mathematical operation accurately. The accuracy of the result depends entirely on the accuracy of the input values for Side A and Side B.
- Where else is the Pythagorean theorem used?
- It's used in navigation (calculating distance), construction, engineering, physics, and many areas of mathematics beyond basic geometry basics.
Related Tools and Internal Resources
- Area Calculator: Calculate the area of various shapes, including triangles.
- Volume Calculator: Find the volume of 3D shapes.
- Pythagorean Theorem Explained: A detailed guide to the theorem used by the hypotenuse of a triangle calculator.
- Geometry Basics: Learn fundamental concepts of geometry.
- Triangle Angle Calculator: Calculate angles of a triangle given sides or other angles.
- Distance Calculator: Find the distance between two points, often using the Pythagorean theorem in a coordinate plane.