# A rectangle has sides of (7 + sqrt2) feet and (7 - sqrt2) feet long. What is the area and the perimeter of the rectangle?

Dec 1, 2015

$P = 28$
$A = 47$

#### Explanation:

Perimeter:

The rectangle's sides are $\left(7 + \sqrt{2}\right) , \left(7 + \sqrt{2}\right) , \left(7 - \sqrt{2}\right) ,$ and $\left(7 - \sqrt{2}\right)$.

To find the perimeter, we add all of these to one another:

$\left(7 + \sqrt{2}\right) + \left(7 + \sqrt{2}\right) + \left(7 - \sqrt{2}\right) + \left(7 - \sqrt{2}\right)$

We can rearrange the order to see that the perimeter equals:

$7 + 7 + 7 + 7 + \sqrt{2} - \sqrt{2} + \sqrt{2} - \sqrt{2}$

Notice that all the square root terms will cancel, and the sevens will add together for a perimeter of 28.

Area:

To find the area of the rectangle, multiply the base length by the height length. We will have to FOIL.

$\left(7 + \sqrt{2}\right) \left(7 - \sqrt{2}\right) = 49 - 7 \sqrt{2} + 7 \sqrt{2} - 2$

Two things: notice that $\sqrt{2} \times - \sqrt{2} = - 2$ and that the $7 \sqrt{2}$ terms will cancel.

This leaves us with an area of $49 - 2$ or 47.