# The diagonals of a square each measure 7 feet. How do you find the area of the square?

Oct 19, 2015

4.95 feet

#### Explanation:

You must use trigonometry.
Since it is a square, and each angle in a square is 90 degrees, the two angles of the triangle the diagonal makes is half of that, 45.

$\sin \theta =$(opposite)/hypoteneus
$\sin 45 = \frac{o}{7}$
$7 \sin 45 = o$
$o = 4.94974746831$

You can use the Pythagorean Theorem

$A = 24.5$ square feet

#### Explanation:

The Pythagorean Theorem says that in a square triangle the square of the hypotenuse (a) is equal to the square of one leg (b) plus the square of the other lag (c):

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

The diagonal of a square form a square triangle with two of the sides (see figure below) and (b) is equal to (c). Therefore, the equation can be written as below:

${a}^{2} = {b}^{2} + {b}^{2}$

${7}^{2} = {b}^{2} + {b}^{2}$

$49 = 2 {b}^{2}$

${b}^{2} = \frac{49}{2}$

$b = \sqrt{\frac{49}{2}}$ feet (side of the square)

The area is the product of two sides

$A = \sqrt{\frac{49}{2}} \times \sqrt{\frac{49}{2}} = \frac{49}{2} = 24.5$ square feet