Hypotenuse Formula
In a rightangled triangle, the longest side is called the hypotenuse, and in this section, we will learn about the hypotenuse formula. The hypotenuse is always the side that is opposite to the right angle in a right triangle. We use Pythagoras theorem to find the hypotenuse, and thus the hypotenuse formula. By Pythagoras theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides (legs).
Here,
c = hypotenuse
a = height (a leg)
b = base (a leg)
What Is Hypotenuse Formula?
From Pythagoras theorem,
(hypotenuse)^{2} = (base)^{2} + (height)^{2}
If we take the square root on both sides, we get the formula of the hypotenuse.
hypotenuse = √ ((base)^{2} + (height)^{2})
(or)
c = √(a^{2} + b^{2})
Solved Examples Using Hypotenuse Formula

Example 1
Solve for the longest side of the given bread slice that is similar to a rightangle triangle. Its height is 12 units and its base is 5 units.
Solution:
To find: The longest side (hypotenuse) of the bread slice.
Its height = 12 units.
Its base = 5 units.
By using the hypotenuse formula,
hypotenuse = √ ((base)^{2} + (height ^{2})
\[ \begin{align} &= \sqrt{5^2+12^2}\\[0.2cm] &= \sqrt{25+144}\\[0.2cm] &= \sqrt{169}\\[0.2cm] &= 13 \end{align}\]
Answer: The longest side of the bread slice = 13 units.

Example 2
The hypotenuse of a right triangle is 5 units. Its base is 3 units. Find its height.
Solution:
To find: The height of the right triangle.
Let us assume it to be 'h'
Its hypotenuse = 5 units.
Its base = 3 units.
By using the hypotenuse formula,
hypotenuse^{2} = (base)^{2} + (height)^{2}
\[ \begin{align} 5^2&= 3^2+h^2\\[0.2cm] 25&= 9+h^2\\[0.2cm] 259&=h^2\\[0.2cm] 16&=h^2\\[0.2cm]4&=h \end{align}\]
Answer: The height of the right triangle = 4 units.