Squares - ACT Math

Card 0 of 297

Question

How much more area does a square with a side of 2r have than a circle with a radius r? Approximate π by using 22/7.

Answer

The area of a circle is given by A = πr2 or 22/7r2

The area of a square is given by A = s2 or (2r)2 = 4r2

Then subtract the area of the circle from the area of the square and get 6/7 square units.

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Question

If a completely fenced-in square-shaped yard requires 140 feet of fence, what is the area, in square feet, of the lot?

Answer

Since the yard is square in shape, we can divide the perimeter(140ft) by 4, giving us 35ft for each side. We then square 35 to give us the area, 1225 feet.

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Question

If the perimeter of a square is 44 centimeters, what is the area of the square in square centimeters?

Answer

Since the square's perimeter is 44, then each side is $\dpi{100} \small \frac{44}{4}=11$ .

Then in order to find the area, use the definition that the

$\dpi{100} \small Area=side^{2}$

$\dpi{100} \small 11^{2}=121$

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Question

Given square , with midpoints on each side connected to form a new, smaller square. How many times bigger is the area of the larger square than the smaller square?

Answer

Assume that the length of each midpoint is 1. This means that the length of each side of the large square is 2, so the area of the larger square is 4 square units. $A=s^{2}$

To find the area of the smaller square, first find the length of each side. Because the length of each midpoint is 1, each side of the smaller square is $\sqrt{2}$ (use either the Pythagorean Theorem or notice that these right trianges are isoceles right trianges, so $s, s, s\sqrt{2}$ can be used).

The area then of the smaller square is 2 square units.

Comparing the area of the two squares, the larger square is 2 times larger than the smaller square.

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Question

Eric has 160 feet of fence for a parking lot he manages. If he is using all of the fencing, what is the area of the lot assuming it is square?

Answer

The area of a square is equal to its length times its width, so we need to figure out how long each side of the parking lot is. Since a square has four sides we calculate each side by dividing its perimeter by four.

$\frac{160}{4}=40$

Each side of the square lot will use 40 feet of fence.

$Area=length \times width$

$40\ ft\times 40\ ft=1600\ ft^{2}$ .

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Question

A square is circumscribed on a circle with a 6 inch radius. What is the area of the square, in square inches?

Answer

We know that the radius of the circle is also half the length of the side of the square; therefore, we also know that the length of each side of the square is 12 inches.

We need to square this number to find the area of the square.

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Question

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

Answer

For a square all the sides are equal, and there are four sides, so divide the perimeter by 4 to determine the side length.

.

Next to find the area of a square, square the side length:

.

Don't forget your units!

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Question

The perimeter of a square is $\textup{in}$ . What is its area?

Answer

The perimeter of a square is very easy to calculate. Since all of the sides are the same in length, it is merely:

From this, you can calculate the area merely by squaring the side's value:

$\textup{in}^2$

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Question

What is the area of a square with a side length of $14\textup{yd}$ ?

Answer

The area of a square is very easy. You merely need to square the length of any given side. That is, the area is defined as:

For our data, this is:

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Question

A square garden is inscribed inside a circular cobblestone path. If the radius of the cobblestone path is feet, what is the area of the garden?

Answer

If a square is inscribed inside a circle, the diameter of the circle is the diagonal of the square. Since we know the radius of the circle is feet, the diameter must be feet. Thus, the diagonal of the square garden is feet.

All squares have congruent sides; thus, the diagonal of a square creates two isosceles right triangles. The ratio of the lengths of the sides of an isosceles right triangle are $x: x: x\sqrt2$ , where $x\sqrt2$ is the hypotenuse. Thus, to find the length of a side of a square from the diagonal, we must divide by $\sqrt2$ .

$s = \frac{90}{\sqrt2} = \frac{90\sqrt2}{2} = 45\sqrt2$

The area of a square is , so if one side is $45\sqrt2$ , our area is

$A = s^2 = (45\sqrt2)^2 = (2025\cdot 2) = 4050$

Thus, the area of our square garden is

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Question

Find the area of a square whose side length is .

Answer

To find area, simply square the side length. Thus,

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Question

Find the area of a square with side length 5.

Answer

To solve, simply use the formula for the area of a square given side length s. Thus,

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Question

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

Answer

$Area = Length \times Length$

$Length = \sqrt{100}=10$

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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:

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

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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:

$6.3662 \div 1.4142 \approx 4.5$

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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 $\pi$ :

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

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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.

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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 $\frac{8} {9} \times 100 %= 88 \frac{8} {9} %$

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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%.

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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,

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

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