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ISEE Upper Level Math : How to find the area of a square

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Example Questions

Example Question #1 : How To Find The Area Of A Square

A square is made into a rectangle by increasing the width by 20% and decreasing the length by 20%. By what percentage has the area of the square changed?

Possible Answers:

increased by 20%

decreased by 10%

decreased by 4%

the area remains the same

Correct answer:

decreased by 4%

Explanation:

The area decreases by 20% of 20%, which is 4%.

The easiest way to see this is to plug in numbers for the sides of the square. If we are using percentages, it is easiest to use factors of 10 or 100. In this case we will say that the square has a side length of 10.

10% of 10 is 1, so 20% is 2. Now we can just increase one of the sides by 2, and decrease another side by 2. So our rectangle has dimensions of 12 x 8 instead of 10 x 10.

The original square had an area of 100, and the new rectangle has an area of 96. So the rectangle is 4 square units smaller, which is 4% smaller than the original square.

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Example Question #2 : How To Find The Area Of A Square

Side shown in the diagram of square ABCD below is equal to 21cm. What is the area of ABCD ?

Possible Answers:

$210\ cm^{2}$

Cannot be determined

$42\ cm^{2}$

$84\ cm^{2}$

$441\ cm^{2}$

Correct answer:

$441\ cm^{2}$

Explanation:

To find the area of a quadrilateral, multiply length times width. In a square, since all sides are equal, is both the length and width.

$a^{2}=21\times21=441$

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Example Question #2 : How To Find The Area Of A Square

If Amy is carpeting her living room, which meaures feet by feet, how many square feet of carpet will she need?

Possible Answers:

$144 ft^{2}$

$120 ft^{2}$

$100 ft^{2}$

$44 ft^{2}$

$22 ft^{2}$

Correct answer:

$120 ft^{2}$

Explanation:

To find the area of the floor, multiply the length of the room by the width (which is the same forumla used to find the area of a square). The equation can be written: $A=l\cdot w$

Substitute feet for and feet for :

$A=10\cdot 12$

Amy will need $120 ft^{2}$ of carpet.

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Example Question #2 : How To Find The Area Of A Square

A rectangle and a square have the same perimeter. The rectangle has length 100 centimeters and width centimeters. Give the area of the square.

Possible Answers:

$4,000 \textrm{ cm}^{2}$

$19,600 \textrm{ cm}^{2}$

$8,000 \textrm{ cm}^{2}$

$4,900 \textrm{ cm}^{2}$

$3,600 \textrm{ cm}^{2}$

Correct answer:

$4,900 \textrm{ cm}^{2}$

Explanation:

The perimeter of the rectangle is

$P = 2L + 2W = 2\cdot100 +2\cdot40 = 200 + 80 = 280$ centimeters.

This is also the perimeter of the square, so divide this by to get its sidelength:

$s = \frac{P}{4} = 280 \div 4 = 70$ centimeters.

The area is the square of this, or $A = s^{2} = 70^{2} = 4,900$ square centimeters.

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Example Question #3 : How To Find The Area Of A Square

Four squares have sidelengths 4 inches, 8 inches, 12 inches, and 16 inches. What is the average of their areas?

Possible Answers:

$200\textrm{ in} ^{2}$

$144\textrm{ in} ^{2}$

$64\textrm{ in} ^{2}$

$100\textrm{ in} ^{2}$

$120\textrm{ in} ^{2}$

Correct answer:

$120\textrm{ in} ^{2}$

Explanation:

The areas of the four squares can be calculated by squaring their sidelengths. Add these areas, then divide by 4:

$\frac{4^{2}+8^{2}+12^{2}+16^{2}}{4} = \frac{16+64+144+256}{4} = \frac{480}{4} = 120$ square inches

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Example Question #14 : Quadrilaterals

Which of the following is equal to the area of a square with sidelength $1 \frac{1}{6}$ yards?

Possible Answers:

$1,764 \textrm{ in}^{2}$

$1,225 \textrm{ in}^{2}$

$2,304 \textrm{ in}^{2}$

$2,025 \textrm{ in}^{2}$

Correct answer:

$1,764 \textrm{ in}^{2}$

Explanation:

Multiply the sidelength by 36 to convert from yards to inches:

$1 \frac{1}{6} \times36 = \frac{7}{6} \times \frac{36}{1} = \frac{7}{1} \times \frac{6}{1} = 42\ inches$

Square this to get the area:

$42^{2} = 1,764$ square inches

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Example Question #4 : How To Find The Area Of A Square

What is the area of a square in which the length of one side is equal to $\sqrt{\frac{1}{4}+\frac{1}{3}}$ ?

Possible Answers:

$\frac{7}{12}$

$\frac{1}{2}$

$\frac{5}{12}$

$\frac{2}{3}$

Correct answer:

$\frac{7}{12}$

Explanation:

The area of a square is equal to the product of one side multiplied by another side. Therefore, the area will be equal to:

$\sqrt{\frac{1}{4}+\frac{1}{3}}\cdot \sqrt{\frac{1}{4}+\frac{1}{3}}=\frac{1}{4}+\frac{1}{3}$

The next step is to convert the fractions being added together to a form in which they have a common denominator. This gives us:

$\frac{3}{12}+\frac{4}{12}=\frac{7}{12}$

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Example Question #17 : Squares

One of the sides of a square on the coordinate plane has its endpoints at the points with coordinates (6, 2) and (-5, 1) . What is the area of this square?

Possible Answers:

122

120

Correct answer:

122

Explanation:

The length of a segment with endpoints (6, 2) and (-5, 1) can be found using the distance formula with $x_{1} = 6$ , $y_{1} = 2$ , $x_{2} = -5$ , $y_{2} = 1$ :

$d = \sqrt{(x_{2}- x_{1})^{2}+(y_{2}-y_{1})^{2}}$

$= \sqrt{(6 - (-5))^{2}+(2-1)^{2}}$

$= \sqrt{11^{2}+1^{2}}$

$= \sqrt{121+1}$

$= \sqrt{122}$

This is the length of one side of the square, so the area is the square of this, or 122.

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Example Question #18 : Squares

One of the sides of a square on the coordinate plane has its endpoint at the points with coordinates (A, B) and (-B, A) , where and are both positive. Give the area of the square in terms of and .

Possible Answers:

$2A^{2} +2 B^{2}$

$2A^{2} +4AB +2 B^{2}$

$2A^{2} -4AB +2 B^{2}$

4AB

Correct answer:

$2A^{2} +2 B^{2}$

Explanation:

The length of a segment with endpoints (A, B) and (-B, A) can be found using the distance formula as follows:

$d = \sqrt{(x_{2}- x_{1})^{2}+(y_{2}-y_{1})^{2}}$

$d = \sqrt{((-B)-A)^{2}+(A-B)^{2}}$

$d = \sqrt{(-B-A)^{2}+(A-B)^{2}}$

$d = \sqrt{(-1)^{2}(B+A)^{2}+(A-B)^{2}}$

$d = \sqrt{1 \cdot (B^{2} +2AB +A^{2})+(A^{2}-2AB+B^{2})}$

$d = \sqrt{ (B^{2} +2AB +A^{2})+(A^{2}-2AB+B^{2})}$

$d = \sqrt{ 2A^{2} +2 B^{2}}$

This is the length of one side of the square, so the area is the square of this, or $2A^{2} +2 B^{2}$ .

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Example Question #19 : Squares

One of the vertices of a square is at the origin. The square has area 13. Which of the following could be the vertex of the square opposite that at the origin?

Possible Answers:

(3, 3)

(4, 2)

(5, 1)

(6,0)

Correct answer:

(5, 1)

Explanation:

Since a square is a rhombus, one way to calculate the area of a square is to take half the square of the length of a diagonal. If we let be the length of each diagonal, then

$\frac{1}{2} d^{2} = A$

$\frac{1}{2} d^{2} = 13$

$d^{2} =26$

$d = \sqrt{26}$

Therefore, we want to choose the point that is $\sqrt{26}$ units from the origin. Using the distance formula, we see that (5, 1) is such a point:

$d = \sqrt{(x_{2}- x_{1})^{2}+(y_{2}-y_{1})^{2}}$

$= \sqrt{(5-0)^{2}+(1-0)^{2}}$

$= \sqrt{5^{2}+1^{2}}$

$= \sqrt{25+1}$

$= \sqrt{26}$

Of the other points:

(6,0) :

$d = \sqrt{(x_{2}- x_{1})^{2}+(y_{2}-y_{1})^{2}}$

$= \sqrt{(6-0)^{2}+(0-0)^{2}}$

$= \sqrt{6^{2}+0^{2}}$

$= \sqrt{36+0}$

$= \sqrt{36 }$

(4, 2) :

$d = \sqrt{(x_{2}- x_{1})^{2}+(y_{2}-y_{1})^{2}}$

$= \sqrt{(4-0)^{2}+(2-0)^{2}}$

$= \sqrt{4^{2}+2^{2}}$

$= \sqrt{16+4}$

$= \sqrt{20}$

(3, 3) :

$d = \sqrt{(x_{2}- x_{1})^{2}+(y_{2}-y_{1})^{2}}$

$= \sqrt{(3-0)^{2}+(3-0)^{2}}$

$= \sqrt{3^{2}+3^{2}}$

$= \sqrt{9+9}$

$= \sqrt{18}$

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : How to find the area of a square

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : How To Find The Area Of A Square

Example Question #2 : How To Find The Area Of A Square

Example Question #2 : How To Find The Area Of A Square

Example Question #2 : How To Find The Area Of A Square

Example Question #3 : How To Find The Area Of A Square

Example Question #14 : Quadrilaterals

Example Question #4 : How To Find The Area Of A Square

Example Question #17 : Squares

Example Question #18 : Squares

Example Question #19 : Squares

All ISEE Upper Level Math Resources

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