### All ACT Math Resources

## 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?

**Possible Answers:**

**Correct answer:**

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

In Square , . Evaluate in terms of .

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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 #27 : Squares

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

**Possible Answers:**

**Correct answer:**

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 #4 : 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)?

**Possible Answers:**

**Correct answer:**

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

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

**Possible Answers:**

**Correct answer:**

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 #2 : 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 ?

**Possible Answers:**

**Correct answer:**

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

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

**Possible Answers:**

**Correct answer:**

To find side length, simply take the square root of the volume. Thus,

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