How to find the length of the side of a square - ACT Math

Card 0 of 72

Question

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

Answer

$Area = Length \times Length$

$Length = \sqrt{100}=10$

Compare your answer with the correct one above

Question

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

Answer

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:

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

Compare your answer with the correct one above

Question

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

Answer

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:

$6.3662 \div 1.4142 \approx 4.5$

Compare your answer with the correct one above

Question

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

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 $\pi$ :

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

Compare your answer with the correct one above

Question

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 ?

Answer

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.

Compare your answer with the correct one above

Question

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

Answer

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:

$\frac{9}{10} (SQ)^{2} = RE \cdot EC$

$\frac{9}{10} (SQ)^{2} = \frac{4}{5} SQ \cdot EC$

$\frac{9}{10} \cdot SQ = \frac{4}{5} \cdot EC$

$\frac{10} {9} \cdot \frac{9}{10} \cdot SQ = \frac{10} {9} \cdot \frac{4}{5} \cdot EC$

$SQ = \frac{8} {9} \cdot EC$

As a percent, $\frac{8} {9}$ of is $\frac{8} {9} \times 100 %= 88 \frac{8} {9} %$

Compare your answer with the correct one above

Question

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

Answer

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%.

Compare your answer with the correct one above

Question

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

Answer

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

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

Compare your answer with the correct one above

Question

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

Answer

$Area = Length \times Length$

$Length = \sqrt{100}=10$

Compare your answer with the correct one above

Question

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

Answer

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:

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

Compare your answer with the correct one above

Question

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

Answer

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:

$6.3662 \div 1.4142 \approx 4.5$

Compare your answer with the correct one above

Question

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

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 $\pi$ :

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

Compare your answer with the correct one above

Question

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 ?

Answer

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.

Compare your answer with the correct one above

Question

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

Answer

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:

$\frac{9}{10} (SQ)^{2} = RE \cdot EC$

$\frac{9}{10} (SQ)^{2} = \frac{4}{5} SQ \cdot EC$

$\frac{9}{10} \cdot SQ = \frac{4}{5} \cdot EC$

$\frac{10} {9} \cdot \frac{9}{10} \cdot SQ = \frac{10} {9} \cdot \frac{4}{5} \cdot EC$

$SQ = \frac{8} {9} \cdot EC$

As a percent, $\frac{8} {9}$ of is $\frac{8} {9} \times 100 %= 88 \frac{8} {9} %$

Compare your answer with the correct one above

Question

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

Answer

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%.

Compare your answer with the correct one above

Question

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

Answer

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

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

Compare your answer with the correct one above

Question

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

Answer

$Area = Length \times Length$

$Length = \sqrt{100}=10$

Compare your answer with the correct one above

Question

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

Answer

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:

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

Compare your answer with the correct one above

Question

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

Answer

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:

$6.3662 \div 1.4142 \approx 4.5$

Compare your answer with the correct one above

Question

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

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 $\pi$ :

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

Compare your answer with the correct one above

Tap the card to reveal the answer