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

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Question

The area of the square is 81. What is the sum of the lengths of three sides of the square?

Answer

A square that has an area of 81 has sides that are the square root of 81 (side2 = area for a square). Thus each of the four sides is 9. The sum of three of these sides is .

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Question

What is the area of a square with perimeter 64 inches?

Answer

The perimeter of a square is four times its sidelength, so a square with perimeter 64 inches has sides with length 16 inches. Use the area formula:

$A = s^{2} = 16^{2} = 256$

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Question

What is the total area of the surface of the cube shown in the above diagram?

Answer

A cube comprises six faces, each of which is a square. To find its total surface area, find the area of one face by squaring its sidelength:

$25 \times 25 = 625 \textrm{ in}^{2}$

Then multiply this by six:

$6 \times 625 = 3,750\textrm{ in}^{2}$

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Question

What is the total area of the surface of the cube shown in the above diagram?

Answer

A cube comprises six faces, each of which is a square. To find its total surface area, find the area of one face by squaring its sidelength:

$3 \frac{1}{2} \times 3 \frac{1}{2} = \frac{7}{2} \times \frac{7}{2} = \frac{49}{4} \textrm{ in}^{2}$

Then multiply this by six:

$6 \times \frac{49}{4} =\frac{6}{1} \times \frac{49}{4} = \frac{3}{1} \times \frac{49}{2} = \frac{147}{2} = 73 \frac{1}{2} \textrm{ in}^{2}$

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Question

What is the total area of the surface of the cube shown in the above diagram?

Answer

A cube comprises six faces, each of which is a square. To find its total surface area, find the area of one face by squaring its sidelength:

$2.6 \times 2.6 = 6.76 \textrm{ cm}^{2}$

Then multiply this by six:

$6 \times 6.76 = 40.56 \textrm{ cm}^{2}$

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Question

What is the total area of the surface of the cube shown in the above diagram?

Answer

A cube comprises six faces, each of which is a square. To find its total surface area, find the area of one face by squaring its sidelength:

$12 \times 12 = 144 \textrm{ cm}^{2}$

Then multiply this by six:

$6 \times 144 = 864\textrm{ cm}^{2}$

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Question

A square is 9 feet long on each side. How many smaller squares, each 3 feet on a side can be cut out of the larger square?

Answer

Each side can be divided into three 3-foot sections. This gives a total of $3\times3=9$ squares. Another way of looking at the problem is that the total area of the large square is 81 and each smaller square has an area of 9. Dividing 81 by 9 gives the correct answer.

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Question

Order the following from least area to greatest area:

Figure A: A square with sides of length 3 feet each.

Figure B: A rectangle with length 30 inches and width 42 inches.

Figure C: A rectangle with length 2 feet and width 4 feet.

Answer

Figure A has area $3 ^{2} = 3 \times 3 = 9$ square feet.

Figure B has dimensions $30 \div12 = 2 \frac{1}{2}$ feet by $42 \div12 = 3 \frac{1}{2}$ feet, so its area is

$2 \frac{1}{2} \times 3 \frac{1}{2} = \frac{5}{2} \times \frac{7}{2} = \frac{35}{4} = 8 \frac{3}{4}$ square feet.

Figure C has area $2 \times 4 = 8$ square feet.

From least area to greatest, the figures rank C, B, A.

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Question

The length of one side of a square is . What is the square's area?

Answer

The area of any quadrilateral is found by multiplying the length by the width. Because a square has four equal sides, the length and width are the same. For the square in this question, the length and width are .

Remember: area is always given in units2 .

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Question

If a square has a side that is 3 yards long, what is the area in square feet?

Answer

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

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

If one side is 3 yards, this means one side is 9 feet since there are 3 feet in a yard.

$3\ \text{yards}\times3\ \text{feet per yard}=9\ \text{feet}$

Since every side is of equal length, you would multiply 9 feet by 9 feet to find the area.

$A=s\times s$

$A=9\times9=81$

This results in 81 square feet, which is the correct answer.

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram, which shows a square. Give the ratio of the area of the yellow region to that of the white region.

Answer

The area of the entire square is the square of the length of a side, or

$80 \times 80 = 6,400$ .

The area of the right triangle is half the product of its legs, or

$\frac{1}{2} \times 30 \times 60 = 900$ .

The area of the yellow region is therefore the difference of the two, or

.

The ratio of the area of the yellow region to that of the white region is

$\frac{5,500}{900} = \frac{5,500 \div 100}{900\div 100} = \frac{55}{9}$ ; that is, 55 to 9.

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Question

Find the area of a square with a width of 13cm.

Answer

To find the area of a square, we will use the following formula:

$A = l \cdot w$

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

Now, we know the width of the square is 13cm. Because it is a square, all sides are equal. Therefore, the length is also 13cm.

So, we can substitute. We get

$A = 13\text{cm} \cdot 13\text{cm}$

$A = 169\text{cm}^2$

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Question

The area of the square is 81. What is the sum of the lengths of three sides of the square?

Answer

A square that has an area of 81 has sides that are the square root of 81 (side2 = area for a square). Thus each of the four sides is 9. The sum of three of these sides is .

Compare your answer with the correct one above

Question

What is the area of a square with perimeter 64 inches?

Answer

The perimeter of a square is four times its sidelength, so a square with perimeter 64 inches has sides with length 16 inches. Use the area formula:

$A = s^{2} = 16^{2} = 256$

Compare your answer with the correct one above

Question

What is the total area of the surface of the cube shown in the above diagram?

Answer

A cube comprises six faces, each of which is a square. To find its total surface area, find the area of one face by squaring its sidelength:

$25 \times 25 = 625 \textrm{ in}^{2}$

Then multiply this by six:

$6 \times 625 = 3,750\textrm{ in}^{2}$

Compare your answer with the correct one above

Question

What is the total area of the surface of the cube shown in the above diagram?

Answer

A cube comprises six faces, each of which is a square. To find its total surface area, find the area of one face by squaring its sidelength:

$3 \frac{1}{2} \times 3 \frac{1}{2} = \frac{7}{2} \times \frac{7}{2} = \frac{49}{4} \textrm{ in}^{2}$

Then multiply this by six:

$6 \times \frac{49}{4} =\frac{6}{1} \times \frac{49}{4} = \frac{3}{1} \times \frac{49}{2} = \frac{147}{2} = 73 \frac{1}{2} \textrm{ in}^{2}$

Compare your answer with the correct one above

Question

What is the total area of the surface of the cube shown in the above diagram?

Answer

A cube comprises six faces, each of which is a square. To find its total surface area, find the area of one face by squaring its sidelength:

$2.6 \times 2.6 = 6.76 \textrm{ cm}^{2}$

Then multiply this by six:

$6 \times 6.76 = 40.56 \textrm{ cm}^{2}$

Compare your answer with the correct one above

Question

What is the total area of the surface of the cube shown in the above diagram?

Answer

A cube comprises six faces, each of which is a square. To find its total surface area, find the area of one face by squaring its sidelength:

$12 \times 12 = 144 \textrm{ cm}^{2}$

Then multiply this by six:

$6 \times 144 = 864\textrm{ cm}^{2}$

Compare your answer with the correct one above

Question

A square is 9 feet long on each side. How many smaller squares, each 3 feet on a side can be cut out of the larger square?

Answer

Each side can be divided into three 3-foot sections. This gives a total of $3\times3=9$ squares. Another way of looking at the problem is that the total area of the large square is 81 and each smaller square has an area of 9. Dividing 81 by 9 gives the correct answer.

Compare your answer with the correct one above

Question

Order the following from least area to greatest area:

Figure A: A square with sides of length 3 feet each.

Figure B: A rectangle with length 30 inches and width 42 inches.

Figure C: A rectangle with length 2 feet and width 4 feet.

Answer

Figure A has area $3 ^{2} = 3 \times 3 = 9$ square feet.

Figure B has dimensions $30 \div12 = 2 \frac{1}{2}$ feet by $42 \div12 = 3 \frac{1}{2}$ feet, so its area is

$2 \frac{1}{2} \times 3 \frac{1}{2} = \frac{5}{2} \times \frac{7}{2} = \frac{35}{4} = 8 \frac{3}{4}$ square feet.

Figure C has area $2 \times 4 = 8$ square feet.

From least area to greatest, the figures rank C, B, A.

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