ACT Math - 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 $\pi$ :

$\frac{16}{\pi } \approx \frac{16}{3.1416 } \approx 5.1$ .

In Square , $SU = \sqrt{2x}$ . Evaluate in terms of .

$\sqrt{x}$

$2\sqrt{x}$

$x\sqrt{2}$

Explanation

If diagonal $\overline{SU}$ of Square is constructed, then $\bigtriangleup SQU$ is a 45-45-90 triangle with hypotenuse $SU = \sqrt{2x}$ . By the 45-45-90 Theorem, the sidelength can be calculated as follows:

$SQ = \frac{SU}{\sqrt{2}} = \frac{\sqrt{2x}}{\sqrt{2}} = \sqrt{\frac{2x}{2}} = \sqrt{x}$ .

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

Explanation

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

$s=\sqrt{V}=\sqrt{144}=12$

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

Explanation

$Area = Length \times Length$

$Length = \sqrt{100}=10$

Thingy

Refer to the above figure, which shows equilateral triangle $\bigtriangleup CDE$ inside Square . Also, $\overline{EF} \perp \overline{AB}$ .

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 $\bigtriangleup CDE$ is

$\frac{s^{2}\sqrt{3}}{4} = \frac{\sqrt{3}}{4} \cdot s^{2} \approx \frac{1.7321}{4} \cdot s^{2} \approx 0.4330 s^{2}$

The area of Square is $s^{2}$ .

By symmetry, $\overline{EF}$ 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:

$\frac{1}{2}\left (s ^{2} - 0.4330 s ^{2} \right ) = 100$

$\frac{1}{2}\left ( 0.5670 s ^{2} \right ) = 100$

$0.2835 s ^{2} = 100$

$s ^{2} \approx 100 \div 0.2835$

$s ^{2} \approx 352.73$

$s \approx\sqrt{ 352.73} \approx 18.7$

Of the five choices, 20 comes closest.

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

$A = s^{2}$ ,

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

$0.88A = 0.88s^{2}$

$\sqrt{0.88A} = \sqrt{0.88s^{2}} = \sqrt{0.88} \cdot \sqrt{s^{2}} \approx 0.94 s$

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

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

Explanation

The diameter of a circle with circumference 20 is

$\frac{20}{\pi } \approx \frac{20 }{3.1416} \approx 6.3662$

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

If diagonal $\overline{SU}$ of Square is constructed, then $\bigtriangleup SQU$ is a 45-45-90 triangle with hypotenuse approximately 6.3662. By the 45-45-90 Theorem, divide this by $\sqrt{2} \approx 1.4142$ to get the sidelength of the square:

$6.3662 \div 1.4142 \approx 4.5$

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

$88 \frac{8} {9} %$

$112\frac{1}{2} %$

$79\frac{1}{81} %$

$126 \frac{9}{16} %$

Explanation

The area of Square is the square of sidelength , or $(SQ)^{2}$ .

The area of Rectangle is $RE \cdot EC$ . Rectangle has area 90% of that of Square , which is $\frac{9}{10} (SQ)^{2}$ ; is 80% of , so $RE = \frac{4}{5}SQ$ . We can set up the following equation: