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ACT Math : Squares

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

The diagonal of a square has a length of 10 inches. What is the perimeter of the square in inches squared?

Possible Answers:

$20\sqrt{2}$

$5\sqrt{2}$

$10\sqrt{2}$

$6\sqrt{2}$

Correct answer:

$20\sqrt{2}$

Explanation:

Using the Pythagorean Theorem, we can find the edge of a side to be √50, by 2a²=10². This can be reduced to 5√2. This can then be multiplied by 4 to find the perimeter.

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

What is the perimeter of a square with an area of ?

Possible Answers:

Correct answer:

Explanation:

1. Find the side lengths:

$Area= (side)^{2}$

$36=(side)^{2}$

side=6

2. Use the side lengths to find the perimeter:

Perimeter= 4(side)

Perimeter=4(6)=24

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

Find the perimeter of a square whose area is .

Possible Answers:

Correct answer:

Explanation:

To solve, you must first find the side length.

$s=\sqrt{A}=\sqrt{16}=4$

Then, you must multiply the side length by 4 since there are 4 sides. Thus,

P=4*4=16

In this case, volume and perimeter were the same numerical value, but this won't always be the case.

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

Find the perimeter of a square with side length .

Possible Answers:

Correct answer:

Explanation:

To find perimeter, simply multiply side length by . Thus,

$P=1\cdot 4=4$

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

The area of a square is $64\textup{ in}^2$ , what is the perimeter of the square?

Possible Answers:

$\textup{36in}$

24in

$\textup{32in}$

$\textup{16in}$

$\textup{Not enough information is provided.}$

Correct answer:

$\textup{32in}$

Explanation:

Since the sides of a square are all the same, the area of a square can be found by $\textup{side}*\textup{side}=64\textup{in}^2.$ Therefore, the side of the square must be $8\textup{in}.$ The perimeter of a square can be found by adding up all of the four sides: $\textup{8in+8in+8in+8in=32in.}$

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Example Question #3 : 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:

Explanation:

$Area = Length \times Length$

100 = (Length)^2

$Length = \sqrt{100}=10$

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Example Question #8 : How To Find The Length Of The Side Of A Square

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

Possible Answers:

$2\sqrt{x}$

$x\sqrt{2}$

$\sqrt{x}$

Correct answer:

$\sqrt{x}$

Explanation:

If diagonal $\overline{SU}$ of Square SQUA 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:

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

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

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

Possible Answers:

4.5

3.2

2.3

6.4

9.0

Correct answer:

4.5

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 SQUA 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:

$6.3662 \div 1.4142 \approx 4.5$

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

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

Possible Answers:

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

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

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

$70 \%$

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

Correct answer:

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

Explanation:

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

The area of Rectangle RECT is $RE \cdot EC$ . Rectangle RECT has area 90% of that of Square SQUA , 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 $\frac{8} {9} \times 100 \%= 88 \frac{8} {9} \%$

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Example Question #1 : 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:

$12 \%$

$3 \%$

$9 \%$

$6 \%$

$15 \%$

Correct answer:

$6 \%$

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 0.88A ; 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%.

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ACT Math : Squares

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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

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

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

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

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

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

Example Question #8 : How To Find The Length Of The Side Of A Square

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

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

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

All ACT Math Resources

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