# The area of a square is 81 square centimeters. What is the length of the diagonal?

Nov 13, 2015

If you note that $81$ is a perfect square, you can say that for a real square shape:

$\sqrt{81} = 9$

Furthermore, since you have a square, the diagonal, which forms a hypotenuse, creates a ${45}^{\circ} - {45}^{\circ} - {90}^{\circ}$ triangle.

So, we would expect the hypotenuse to be $9 \sqrt{2}$ since the general relationship for this special type of triangle is:

• $a = n$
• $b = n$
• $c = n \sqrt{2}$

Let's show that $c = 9 \sqrt{2}$ using the Pythagorean Theorem.

$c = \sqrt{{a}^{2} + {b}^{2}}$

$= \sqrt{{9}^{2} + {9}^{2}}$

$= \sqrt{81 + 81}$

$= \sqrt{2 \cdot 81}$

= color(blue)(9sqrt2 " cm"