Hypotenuse Calculator

How to Calculate the Hypotenuse of a Right Triangle

The hypotenuse is the longest side of a right triangle, opposite the right angle. There are two main methods to calculate the hypotenuse: using the Pythagorean theorem or trigonometry.

1. Using the Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.

\[ c^2 = a^2 + b^2 \]

Where:

• \( c \) is the length of the hypotenuse
• \( a \) and \( b \) are the lengths of the other two sides

Calculation Steps:

1. Square the lengths of the two known sides.
2. Add these squared values together.
3. Take the square root of the sum to find the hypotenuse length.

Example:

For a right triangle with sides 3 and 4 units:

1. \( 3^2 = 9 \) and \( 4^2 = 16 \)
2. \( 9 + 16 = 25 \)
3. \( c = \sqrt{25} = 5 \)

The hypotenuse is 5 units long.

2. Using Trigonometry

If you know one side length and one angle (other than the right angle), you can use trigonometric ratios to find the hypotenuse.

\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]

Where \( \theta \) is the known angle.

Calculation Steps:

1. Identify the known side (opposite or adjacent to the known angle).
2. Use the appropriate trigonometric ratio (sine for opposite, cosine for adjacent).
3. Rearrange the formula to solve for the hypotenuse.
4. Calculate the result.

Example:

For a right triangle with a side of 10 units and an angle of 30°:

1. The known side (10) is opposite to the 30° angle.
2. Use sine: \( \sin(30°) = \frac{10}{\text{hypotenuse}} \)
3. Rearrange: \( \text{hypotenuse} = \frac{10}{\sin(30°)} \)
4. Calculate: \( \text{hypotenuse} = \frac{10}{0.5} = 20 \)

The hypotenuse is 20 units long.

Visual Representation

Here's a visual representation of a right triangle with its hypotenuse:

This diagram illustrates a right triangle where 'c' represents the hypotenuse, 'a' and 'b' are the other two sides, and 'θ' is an angle used in the trigonometric method.