The formula for calculating the hypotenuse of a right-angled triangle is given by

In a right-angled triangle, the hypotenuse is the longest side, positioned opposite the right angle. This formula, $c = \sqrt{a^2 + b^2}$, is derived from Pythagoras’ Theorem, which states that in a right-angled triangle, the square of the hypotenuse $c$ is equal to the sum of the squares of the other two sides, $a$ and $b$. This relationship can be expressed as

To isolate the hypotenuse, we take the square root of both sides, leading to the formula

For instance, consider a right-angled triangle where one side $a$ measures $3$ units and the other side $b$ measures $4$ units. The steps to calculate the hypotenuse $c$ are as follows:

1. Square each side:
   • $3^2 = 9$
   • $4^2 = 16$.
2. Sum the squares:
   • $9 + 16 = 25$.
3. Take the square root of the total:
   • $\sqrt{25} = 5$.

Thus, the hypotenuse $c$ is $5$ units long. This formula is crucial for solving various problems in geometry and trigonometry, as it clarifies the relationship between the sides of a right-angled triangle. It is important to note that this theorem is applicable only to right-angled triangles, where one angle measures exactly $90$ degrees.