# ACT Math : How to find the length of the side of a square

## Example Questions

### Example Question #1 : How To Find The Length Of The Side Of A Square

If the area of the square is 100 square units, what is, in units, the length of one side of the square?

Explanation:

### Example Question #2 : How To Find The Length Of The Side Of A Square

In Square . Evaluate  in terms of .

Explanation:

If diagonal  of Square  is constructed, then  is a 45-45-90 triangle with hypotenuse . By the 45-45-90 Theorem, the sidelength  can be calculated as follows:

.

### Example Question #3 : How To Find The Length Of The Side Of A Square

The circle that circumscribes Square  has circumference 20. To the nearest tenth, evaluate .

Explanation:

The diameter of a circle with circumference 20 is

The diameter of a circle that circumscribes a square is equal to the length of the diagonals of the square.

If diagonal  of Square  is constructed, then  is a 45-45-90 triangle with hypotenuse approximately 6.3662. By the 45-45-90 Theorem, divide this by  to get the sidelength of the square:

### Example Question #9 : How To Find The Length Of The Side Of A Square

Rectangle  has area 90% of that of Square , and  is 80% of . What percent of  is ?

Explanation:

The area of Square  is the square of sidelength , or .

The area of Rectangle  is . Rectangle  has area 90% of that of Square , which is ;   is 80% of , so . We can set up the following equation:

As a percent,  of  is

### Example Question #3 : How To Find The Length Of The Side Of A Square

Reducing the area of a square by 12% has the effect of reducing its sidelength by what percent (hearest whole percent)?

Explanation:

The area of the square was originally

being the sidelength.

Reducing the area by 12% means that the new area is 88% of the original area, or ; the square root of this is the new sidelength, so

Each side of the new square will measure 94% of the length of the old measure - a reduction by 6%.

### Example Question #4 : How To Find The Length Of The Side Of A Square

The circle inscribed inside Square  has circumference 16. To the nearest tenth, evaluate .

Explanation:

The diameter of a circle that is inscribed inside a square is equal to its sidelength , so all we need to do is find the diameter of the circle - which is circumference 16 divided by :

.

### Example Question #5 : How To Find The Length Of The Side Of A Square

Refer to the above figure, which shows equilateral triangle  inside Square . Also, .

Quadrilateral  has area 100. Which of these choices comes closest to ?

Explanation:

Let , the sidelength shared by the square and the equilateral triangle.

The area of  is

The area of Square  is .

By symmetry,  bisects the portion of the square not in the triangle, so the area of Quadrilateral  is half the difference of those of the square and the triangle. Since the area of Quadrilateral is 100, we can set up an equation:

Of the five choices, 20 comes closest.

### Example Question #1 : How To Find The Length Of The Side Of A Square

Find the length of the side of a square given its area is .