How to find the area of a square - ISEE Middle Level Quantitative Reasoning

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Question

Calvin is remodeling his room. He used $\small 32$ feet of molding to put molding around all four walls. Now he wants to paint three of the walls. Each wall is the same width and is $\small 8$ feet tall. If one can of paint covers $\small \small \small 24$ square feet, how many cans of paint will he need to paint three walls.

Answer

When Calvin put up $\small 32$ feet of molding, he figured out the perimeter of the room was $\small 32$ feet. Since he knows that all four walls are the same width, he can use the equation $\small 4s=P$ to determine the length of each side by plugging $\small 32$ in for $\small P$ and solving for $\small s$ .

$\small 4s=32$

In order to solve for $\small s$ , Calvin must divide both sides by four.

The left-hand side simplifies to:

$\small \frac{4s}{4}=s$

The right-hand side simplifies to:

$\small \frac{32}{2}=8$

Now, Calvin knows the width of each room is $\small 8$ feet. Next he must find the area of each wall. To do this, he must multiply the width by the height because the area of a rectangle is found using the equation $\small \small A=l\cdot w$ . Since Calvin now knows that the width of each wall is $\small 8$ feet and that the height of each wall is also $\small 8$ feet, he can multiply the two together to find the area.

$\small 8\cdot 8=64$

Since Calvin wants to find how much paint he needs to cover three walls, he must first find out how many square feet he is covering. If one wall is $\small 64$ square feet, he must multiply that by $\small 3$ .

$\small 64\cdot 3=192$

Calvin is painting $\small 192$ square feet. If one can of paint covers 24 square feet, he must divide the total space ( $\small 192$ square feet) by $\small 24$ .

$\small \frac{192}{24}=8$

Calvin will need $\small 8$ cans of paint.

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Question

Which is the greater quantity?

(a) The surface area of a cube with volume $512 \textrm{ cm} ^{3}$

(b) The surface area of a cube with sidelength $90 \textrm{ mm}$

Answer

We can actually solve this by comparing volumes; the cube with the greater volume has the greater sidelength and, subsequently, the greater surface area.

The volume of the cube in (b) is the cube of 90 millimeters, or 9 centimeters. This is $9 ^{3} = 729 \textrm{ cm}^{3}$ , which is greater than $512\ cm^{3}$ . The cube in (b) has the greater volume, sidelength, and, most importantly, surface area.

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Question

Each side of a square is units long. Which is the greater quantity?

(A) The area of the square

(B)

Answer

The area of a square is the square of its side length:

Using the side length from the question:

$A=\left ( 5t\right ) ^{2} = 5 ^{2} \cdot t ^{2} = 25t ^{2}$

However, it is impossible to tell with certainty which of $25t ^{2}$ and is greater.

For example, if ,

$25t ^{2} = 25 \cdot 2^{2} = 25 \cdot4 = 100$

and

$25t = 25 \cdot 2 = 50$

so $25t ^{2} > 25t$ if .

But if $t = \frac{1}{5}$ ,

$25t ^{2}=25 \cdot \left ( \frac{1}{5} \right )^{2} = 25 \cdot \frac{1}{25} = 1$

and

$25t =25 \cdot \frac{1}{5} = 5$

so $25t ^{2} < 25t$ if $t = \frac{1}{5}$ .

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Question

The sum of the lengths of three sides of a square is one yard. Give its area in square inches.

Answer

A square has four sides of the same length.

One yard is equal to 36 inches, so each side of the square has length

$36 \div 3 = 12$ inches.

Its area is the square of the sidelength, or

$12^{2} = 12 \times 12 = 144$ square inches.

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Question

The sum of the lengths of three sides of a square is 3,900 centimeters. Give its area in square meters.

Answer

100 centimeters are equal to one meter, so 3,900 centimeters are equal to

$3,900 \div 100 = 39$ meters.

A square has four sides of the same length. Since the sum of the lengths of three of the congruent sides is 3,900 centimeters, or 39 meters, each side measures

$39 \div 3 = 13$ meters.

The area of the square is the square of the sidelength, or

$13^{2} =13 \times 13 = 169$ square meters.

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Question

A square has a side with a length of 5. What is the area of the square?

Answer

The area formula for a square is length times width. Keep in mind that all of a square's sides are equal.

$A=l\times w=s\times s$

So, if one side of a square equals 5, all of the other sides must also equal 5. You will find the area of the square by multiplying two of its sides:

$A=s\times s=5 \times 5 = 25$

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Question

One square mile is equivalent to 640 acres. Which of the following is the greater quantity?

(a) The area of a square plot of land whose perimeter measures one mile

(b) 160 acres

Answer

A square plot of land with perimeter one mile has as its sidelength one fourth of this, or $\frac{1}{4}$ mile; its area is the square of this, or

$\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$ square miles.

One square mile is equivalent to 640 acres, so $\frac{1}{16}$ square miles is equivalent to

$\frac{1}{16} \times 640 = \frac{640}{16} = 40$ acres.

This makes (b) greater.

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Question

One square kilometer is equal to 100 hectares.

Which is the greater quantity?

(a) The area of a rectangular plot of land 500 meters in length and 200 meters in width

(b) One hectare

Answer

One kilometer is equal to 1,000 meters, so divide each dimension of the plot in meters by 1,000 to convert to kilometers:

$500 \div 1,000= 0.5$ kilometers

$200 \div 1,000 = 0.2$ kilometers

Multiply the dimensions to get the area in square kilometers:

$0.5 \times 0.2 = 0.1$ square kilometers

Since one square kilometer is equal to 100 hectares, multiply this by 100 to convert to hectares:

$0.1 \times 100 = 10$ hectares

This makes (a) the greater.

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Question

What is the area of a square with a side length of 4?

Answer

The area of a square is represented by the equation $\dpi{100} Area = side^{2}$ .

Therefore the area of this square is $\dpi{100} 4^{2}=16$ .

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Question

Square A has sides measuring 5 meters. A second square, Square B, has sides that are 2 meters longer than the sides of Square A. What is the difference in area of Square A and Square B?

Answer

The area of Square A is 5 * 5, or 25 m2.

Since each of Square B's sides is 2 meters longer, the sides measure 7 meters. Therefore, the area of square B is 49 m2.

Subtract to find the difference in areas: $49-25=24\ m^2$

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Question

Which of the following is equal to the area of a square with perimeter 8 meters?

Answer

The sidelength of a square is one-fourth its perimeter, so the sidelength here is one-fourth of 8, or 2, meters. One meter is equal to 100 centimeters, so the sidelength is 200 centimeters. Square this to get the area:

$200 ^{2} = 200 \times 200 = 40,000$ square centimeters.

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Question

Which of the following is the area of a square with perimeter 7 feet?

Answer

Convert 7 feet to inches by multiplying by 12: $7 \times 12 = 84$ inches.

The sidelength of a square is one-fourth its perimeter, so the sidelength here is one-fourth of 84 inches. This is $84 \div 4 = 21$ inches.

Square this to get the area:

$21 ^{2} = 21 \times 21 = 441$ square inches

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Question

The ratio of the length of a side of one square to the length of the side of another square is $\frac{a}{b}$ . Give the ratio of the area of the second square to the area of the first square.

Answer

The area of a square can be found as follows:

Where:

$x=Side\ Length$

So we can write:

$\frac{Area\ 2}{Area\ 1}=\frac{b^2}{a^2}$

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Question

What is the area of a square if the length of one side is $\sqrt{3}$ ?

Answer

The area of a square is found by multiplying one side by itself.

$A=s\times s=s^2$

We are given the side length, allowing us to solve.

$s=\sqrt{3}$

$A=(\sqrt{3})^2=3$

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Question

A square has an area of 36, and each side is equal to What is the value of ?

Answer

Each side of a square is equal to the square root of the area. The square root of 36 is 6, so each side is 6. Thus, 3x (being a side of the square is equal to 6. This means that x is equal to 2 because 3 times 2 is 6.

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Question

Find the area of a square with side length 10.

Answer

To find the area of a square, simply use the formula. Thus,

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Question

The area of a square is ?

Answer

The area of a square is the side length squared not the side length times .

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Question

If you are given one side length of a square, you can find the area with that information.

Answer

To find the area of a square, you multiple . But with a square all the sides are equal so the equation really is or the side length squared. Since you are given the side length, you can find the area.

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Question

A new building in New York City will have a footprint that is a perfect square, taking up a complete city block. If one city block is of a mile, what is the area taken up by the building?

Answer

A new building in New York City will have a footprint that is a perfect square, taking up a complete city block. If one city block is of a mile, what is the area taken up by the building?

Find the area of a square by raising the side length to the second power:

$A=s^2=\frac{1}{8}^2=\frac{1}{8}*\frac{1}{8}=\frac{1}{64}mi^2$

So our area is:

$\frac{1}{64}mi^2$

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Question

Use the following to answer the question.

Find the area of the square.

Answer

When finding the area of a square, we use the following formula:

$\text{area of square} = l \cdot w$

where l is the length and w is the width of the square.

Now, we know that a square has 4 equal sides. So given the square

we know the length is 6ft and the width is 6ft. So, we can substitute into the formula.

$\text{area of square} = 6\text{ft} \cdot 6\text{ft}$

$\text{area of square} = 36\text{ft}^2$

Note that the label is also squared. This is because when we multiply, we are also multiplying the label.

$\text{ft} \cdot \text{ft} = \text{ft}^2$

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