How to find the length of the side of a square - Basic Geometry

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Question

A square has an area of $40\ cm^{2}$ , what is the length of its side?

Answer

The sides can be found by taking the square root of the area.

$(Area)^{1/2}=s$ , where = side.

$(40)^{1/2}=6.3$ .

So the length of a side is 6.3 cm.

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Question

The perimeter of a square is 16. Find the length of each side of this square.

Answer

First, know that all the side lengths of a square are equal. Second, know that the sum of all 4 side lengths gives us the perimeter. Thus, the square perimeter of 16 is written as

where S is the side length of a square. Solve for this S

$S=\frac{16}{4}=4$

So the length of each side of this square is 4.

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Question

The area of the square shown below is 36 square inches. What is the length of one of the sides?

Answer

The area of any quadrilateral can be determined by multiplying the length of its base by its height.

Since we know the shape here is square, we know that all sides are of equal length. From this we can work backwards by taking the square root of the area to find the length of one side.

$\sqrt{36}=6$

The length of one (and each) side of this square is 6 inches.

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Question

A playground is enclosed by a square fence. The area of the playground is . The perimeter of the fence is . What is the length of one side of the fence?

Answer

We will have two formulas to help us solve this problem, the area and perimeter of a square.

The area of a square is:

$\ast$ ,

where length of the square and width of the square.

The perimeter of a square is:

Plugging in our values, we have:

$\ast$

Since all sides of a square have the same value, we can replace all and with (side). Our equations become:

Therefore, .

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Question

A square has one side of length , what is the length of the opposite side?

Answer

One of the necessary conditions of a square is that all sides be of equal length. Therefore, because we are given the length of one side we know the length of all sides and that includes the length of the opposite side. Since the length of one of the sides is 4 we can conclude that all of the sides are 4, meaning the opposite side has a length of 4.

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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

The perimeter of a square is half its area. What is the length of one side of the square?

Answer

We begin by recalling the formulas for the perimeter and area of a square respectively.

Using these formulas and the fact that the perimeter is half the area, we can create an equation.

$4s=\frac{1}{2}(s^2)$

We can multiply both sides by 2 to eliminate the fraction.

To get one side of the equation equal to zero, we will move everything to the right side.

Next we can factor.

Setting each factor equal to zero provides two potential solutions.

or

However, since a square cannot have a side of length 0, 8 is our only answer.

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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

The perimeter of a square is . What is the length of one side of the square?

Answer

Recall how to find the perimeter of a square:

$\text{Perimeter}=4(\text{side length})$

By dividing both sides by , we can write the following:

$\text{Side length}=\frac{\text{Perimeter}}{4}$

For the square in question,

$\text{Side length}=\frac{96}{4}$

$\text{Side length}=24$

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Question

The perimeter of a square is . What is the length of one side of the square?

Answer

Recall how to find the perimeter of a square:

$\text{Perimeter}=4(\text{side length})$

By dividing both sides by , we can write the following:

$\text{Side length}=\frac{\text{Perimeter}}{4}$

For the square in question,

$\text{Side length}=\frac{444}{4}$

$\text{Side length}=111$

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Question

The perimeter of a square is . What is the length of one side of the square?

Answer

Recall how to find the perimeter of a square:

$\text{Perimeter}=4(\text{side length})$

By dividing both sides by , we can write the following:

$\text{Side length}=\frac{\text{Perimeter}}{4}$

For the square in question,

$\text{Side length}=\frac{10}{4}$

$\text{Side length}=2.5$

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Question

The perimeter of a square is . What is the length of one side of the square?

Answer

Recall how to find the perimeter of a square:

$\text{Perimeter}=4(\text{side length})$

By dividing both sides by , we can write the following:

$\text{Side length}=\frac{\text{Perimeter}}{4}$

For the square in question,

$\text{Side length}=\frac{0.8}{4}$

$\text{Side length}=0.2$

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Question

The perimeter of a square is . What is the length of one side of the square?

Answer

Recall how to find the perimeter of a square:

$\text{Perimeter}=4(\text{side length})$

By dividing both sides by , we can write the following:

$\text{Side length}=\frac{\text{Perimeter}}{4}$

For the square in question,

$\text{Side length}=\frac{0.12}{4}$

$\text{Side length}=0.03$

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Question

The perimeter of a square is . What is the length of one side of the square?

Answer

Recall how to find the perimeter of a square:

$\text{Perimeter}=4(\text{side length})$

By dividing both sides by , we can write the following:

$\text{Side length}=\frac{\text{Perimeter}}{4}$

For the square in question,

$\text{Side length}=\frac{0.4}{4}$

$\text{Side length}=0.1$

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Question

The perimeter of a square is . What is the length of one side of the square?

Answer

Recall how to find the perimeter of a square:

$\text{Perimeter}=4(\text{side length})$

By dividing both sides by , we can write the following:

$\text{Side length}=\frac{\text{Perimeter}}{4}$

For the square in question,

$\text{Side length}=\frac{0.16}{4}$

$\text{Side length}=0.04$

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