# The diagonal of a square is sqrt 84. What is the area of the square.

*print*Print*list*Cite

### 3 Answers

The length of the diagonal of a square with side S is equal to `sqrt(S^2 + S^2)` = `S*sqrt 2` .

If the diagonal of a square is D, the length of the side is `D/sqrt 2`

A square has a diagonal of length `sqrt 84` . The area of the square is `(sqrt 84)^2/2 = 84/2 = 42`

**The area of the square is 42 square units.**

The diagonal of a square is the hypotenuse of a 45-45-90 triangle.

In a 45-45-90 triangle the hypotenuse is the leg multiplied by `sqrt(2).`

Therefore, if given the hypotenuse, divide by `sqrt(2)` to find the length of the leg.

`sqrt(84)/sqrt(2) =sqrt(42)`

Therefore each side of the square is `sqrt(42).`

To find the area of the square: `A=s^2.`

`A = (sqrt(42))^2`

**Therefore the area of the square is: `42 u^2` **

Another way to look at this is to use the formula for the area of a rhombus. A rhombus is a quadrilateral with 4 equal sides. If the length of the diagonals is D1 and D2, the area of the rhombus is `(D1*D2)/2` .

A square is a rhombus and as the adjacent sides are perpendicular to each other, the diagonals are equal in length.

The area of a square with diagonal `sqrt 84` is : `(sqrt 84*sqrt 84)/2 = 84/2 = 41`

The area of the square is 41.