Squares - ACT Math

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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

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

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

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

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

The perimeter of a square is 48. What is the length of its diagonal?

Answer

Perimeter = side * 4

48 = side * 4

Side = 12

We can break up the square into two equal right triangles. The diagonal of the sqaure is then the hypotenuse of these two triangles.

Therefore, we can use the Pythagorean Theorem to solve for the diagonal:

$\sqrt{144+144}=\sqrt{hypotenuse^2}$

$hypotenuse=\sqrt{2\times 144}=\sqrt{2\times 12\times 12}=12\sqrt{2}$

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Question

The perimeter of a square is units. How many units long is the diagonal of the square?

Answer

From the perimeter, we can find the length of each side of the square. The side lengths of a square are equal by definition therefore, the perimeter can be rewritten as,

Then we use the Pythagorean Theorme to find the diagonal, which is the hypotenuse of a right triangle with each leg being a side of the square.

$c=\sqrt{49+49}=\sqrt{98}=\sqrt{2\cdot49}=7\sqrt2$

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Question

What is the area of a square with a diagonal of length ? Round to the nearest hundredth.

Answer

Based on the data provided, the square could be drawn like this:

Based on this, you can use the Pythagorean theorem to find :

or

From this, you know that

Since the area is equal to , this is your answer!

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Question

Find the diagonal of a square with side length .

Answer

To solve, simply realize the triangle that is made by sides and the diagonal is an isoceles right triangle. Thus, the hypotenuse is side length time square root of . Thus,

$s\sqrt2=3\sqrt2$

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Question

Find the length of a diagonal of a square whose side length is .

Answer

To solve, simply remember that the diagonal forms an isoceles right triangle. Thus, the diagonal is:

$x\sqrt2=1*\sqrt2=\sqrt2$

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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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