Find The Unknown Length In The Right Triangle Calculator

Use this calculator to find the length of the unknown side (a, b, or c) of a right-angled triangle using the Pythagorean theorem.

Calculator

Summary of Values

Side	Length	Length Squared
a	?	?
b	?	?
c (Hypotenuse)	?	?

Table showing the lengths of the sides and their squares.

What is a Find the Unknown Length in the Right Triangle Calculator?

A "find the unknown length in the right triangle calculator" is a tool that uses the Pythagorean theorem to determine the length of one side of a right-angled triangle when the lengths of the other two sides are known. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle, denoted as 'c') is equal to the sum of the squares of the lengths of the other two sides (legs, denoted as 'a' and 'b'). The formula is a² + b² = c².

This calculator is useful for students, engineers, architects, builders, and anyone needing to solve for a missing side in a right triangle. Whether you need to find the hypotenuse given the two legs, or one leg given the other leg and the hypotenuse, this find the unknown length in the right triangle calculator simplifies the process.

Common misconceptions include applying the theorem to non-right triangles or incorrectly identifying the hypotenuse. The hypotenuse is always the longest side and is opposite the 90-degree angle.

Find the Unknown Length in the Right Triangle Calculator: Formula and Mathematical Explanation

The core of the find the unknown length in the right triangle calculator is the Pythagorean theorem:

a² + b² = c²

Where:

• a and b are the lengths of the two shorter sides (legs) of the right triangle.
• c is the length of the longest side, the hypotenuse.

Depending on which side is unknown, we rearrange the formula:

• If c (hypotenuse) is unknown: c = √(a² + b²)
• If a is unknown: a = √(c² – b²) (Requires c > b)
• If b is unknown: b = √(c² – a²) (Requires c > a)

Variable	Meaning	Unit	Typical Range
a	Length of one leg	Length (e.g., cm, m, inches, feet)	Positive number
b	Length of the other leg	Length (e.g., cm, m, inches, feet)	Positive number
c	Length of the hypotenuse	Length (e.g., cm, m, inches, feet)	Positive number, c > a and c > b

Variables used in the Pythagorean theorem.

Practical Examples (Real-World Use Cases)

Let's see how our find the unknown length in the right triangle calculator works with examples.

Example 1: Finding the Hypotenuse

Imagine a ladder leaning against a wall. The base of the ladder is 3 meters away from the wall (side b), and the ladder reaches 4 meters up the wall (side a). We want to find the length of the ladder (hypotenuse c).

• Side a = 4 m
• Side b = 3 m
• Unknown side = c

Using the formula c = √(a² + b²) = √(4² + 3²) = √(16 + 9) = √25 = 5 meters. The ladder is 5 meters long.

Example 2: Finding a Leg

A TV screen is advertised as being 50 inches (diagonal, which is the hypotenuse c). The height of the TV is 30 inches (side a). What is the width of the TV (side b)?

• Hypotenuse c = 50 inches
• Side a = 30 inches
• Unknown side = b

Using the formula b = √(c² – a²) = √(50² – 30²) = √(2500 – 900) = √1600 = 40 inches. The width of the TV is 40 inches. You can use the find the unknown length in the right triangle calculator above to verify this.

How to Use This Find the Unknown Length in the Right Triangle Calculator

1. Select the Unknown Side: Use the dropdown menu ("Which side is unknown?") to select whether you want to calculate side 'a', side 'b', or the hypotenuse 'c'.
2. Enter Known Lengths: Input the lengths of the two known sides into the corresponding input fields. The field for the unknown side will be disabled. Ensure you enter positive values.
3. View Results: The calculator will automatically update and display the length of the unknown side in the "Results" section as you type. It will also show intermediate calculations and update the table and diagram.
4. Check Diagram and Table: The visual diagram and the summary table will update with the values you entered and the calculated result.
5. Reset or Copy: Use the "Reset" button to clear the inputs and start over with default values. Use "Copy Results" to copy the main result and intermediate values to your clipboard.

When finding a leg (a or b), ensure the hypotenuse (c) is longer than the known leg. The find the unknown length in the right triangle calculator will show an error if this condition isn't met.

Key Factors That Affect Find the Unknown Length in the Right Triangle Calculator Results

• Accuracy of Input Values: The most crucial factor is the precision of the lengths of the known sides. Small errors in input can lead to inaccuracies in the calculated length.
• Correct Identification of Sides: Ensure you correctly identify sides 'a', 'b', and 'c' (hypotenuse). 'c' is always opposite the right angle and is the longest side.
• Units of Measurement: The units of the calculated length will be the same as the units used for the input lengths. Be consistent (e.g., all cm or all inches).
• Is it a Right Triangle?: The Pythagorean theorem and this calculator only apply to right-angled triangles. If the triangle is not a right triangle, the results will be incorrect.
• Rounding: The calculator may round the result to a certain number of decimal places. For high-precision needs, be aware of the rounding used.
• Valid Inputs: Side lengths must be positive numbers. When calculating a leg, the hypotenuse must be longer than the other known leg (c > a and c > b), otherwise, the value under the square root will be negative, which is not possible for real-world lengths. Our find the unknown length in the right triangle calculator checks for this.

Frequently Asked Questions (FAQ)

What is the Pythagorean theorem?

The Pythagorean theorem is a fundamental relation in Euclidean 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.

Can I use this calculator for any triangle?

No, this find the unknown length in the right triangle calculator is specifically for right-angled triangles. For other triangles, you might need the Law of Sines or the Law of Cosines (see our Law of Sines Calculator).

What if I enter a negative number?

Side lengths cannot be negative. The calculator will show an error or prevent calculation if negative numbers are entered for lengths.

What if the hypotenuse I enter is shorter than a leg?

If you are calculating a leg (a or b) and enter a hypotenuse (c) value that is less than or equal to the other known leg, the calculator will indicate an error because it's mathematically impossible for a leg to be longer than or equal to the hypotenuse in a right triangle, leading to a negative value under the square root.

What units can I use?

You can use any unit of length (cm, m, inches, feet, etc.), as long as you are consistent for all sides. The result will be in the same unit. Our unit converter can help.

How accurate is this find the unknown length in the right triangle calculator?

The calculator performs the mathematical operations accurately based on the formula. The accuracy of the result depends on the accuracy of your input values and the rounding applied by the display.

What is the hypotenuse?

The hypotenuse is the longest side of a right-angled triangle, and it is always the side opposite the right angle (90-degree angle).

Why is it called the Pythagorean theorem?

It is named after the ancient Greek mathematician Pythagoras, who is credited with its first proof, although the theorem was known to earlier civilizations. More about triangles can be found on our Triangle Area Calculator page.