GMAT Math : Calculating the area of a square

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Example Questions

Example Question #11 : Calculating The Area Of A Square

Squares

Note: Figure NOT drawn to scale

Refer to the above figure, which shows Square \displaystyle ABCD and Square \displaystyle WXYZ. The ratio of \displaystyle BX to \displaystyle XC is 13 to 2. What is the ratio of the area of Square \displaystyle ABCD to that of Square \displaystyle WXYZ?

Possible Answers:

\displaystyle 225:221

\displaystyle 225:169

\displaystyle 15:13

\displaystyle 225:173

\displaystyle 17:15

Correct answer:

\displaystyle 225:173

Explanation:

To make this easier, assume that \displaystyle BX = 13 and \displaystyle BW=CX = 2 - the reasoning generalizes. Then Square \displaystyle ABCD has sidelength 15 and area \displaystyle 15^{2}= 225. The sidelength of Square \displaystyle WXYZ, each side being a hypotenuse of a right triangle with legs 2 and 13, is 

\displaystyle \sqrt{2^{2}+13^{2}}= \sqrt{4+169} = \sqrt{173}.

The square of this, 173, is the area of Square \displaystyle WXYZ.

The ratio is \displaystyle 225:173.

Example Question #171 : Geometry

Squares

Note: Figure NOT drawn to scale

Refer to the above figure, which shows Square \displaystyle ABCD and Square \displaystyle WXYZ.  The ratio of \displaystyle BX to \displaystyle XC is 7 to 1.

Which of these responses comes closest to what percent the area of Square \displaystyle WXYZ is of that of Square \displaystyle ABCD?

 

Possible Answers:

\displaystyle 75 \%

\displaystyle 90 \%

\displaystyle 85 \%

\displaystyle 70 \%

\displaystyle 80 \%

Correct answer:

\displaystyle 80 \%

Explanation:

To make this easier, assume that \displaystyle BX = 7 and \displaystyle BW = XC = 1; the results generalize. 

Each side of Square \displaystyle ABCD has length 8, so the area of Square \displaystyle ABCD is 64. 

Each of the four right triangles has legs 7 and 1, so each has area \displaystyle \frac{1}{2} \cdot 1 \cdot 7 = 3\frac{1}{2}; Square \displaystyle WXYZ has area four times this subtracted from the area of Square \displaystyle ABCD, or

\displaystyle 64 - 4 \cdot 3 \frac{1}{2} = 50.

The area of Square \displaystyle WXYZ is

\displaystyle \frac{50}{64} \times 100 \% = 78 \frac{1}{8} \%

of that of Square \displaystyle ABCD.

Of the five choices, 80% comes closest.

Example Question #61 : Quadrilaterals

The perimeter of a square is the same as the circumference of a circle with area 100. What is the area of the square?

Possible Answers:

\displaystyle 25 \pi

\displaystyle \frac{25 \pi^{2}}{4}

\displaystyle \frac{200 \sqrt{\pi}}{\pi}

\displaystyle 50 \sqrt{\pi}

\displaystyle \frac{100 }{\pi}

Correct answer:

\displaystyle 25 \pi

Explanation:

The formula for the area of a circle is

\displaystyle A = \pi r^{2}.

If the area is 100, the radius is as follows:

\displaystyle \pi r^{2} = 100

\displaystyle r^{2} = \frac{100}{\pi}

\displaystyle r = \sqrt{\frac{100}{\pi}}= \frac{10}{\sqrt{\pi}}

The circle has circumference \displaystyle 2 \pi times its radius, or

\displaystyle C= \frac{10}{\sqrt{\pi}} \cdot 2 \pi = 20 \sqrt{\pi}

This is also the perimeter of the square, so the sidelength of the square is one-fourth this, or

\displaystyle S = \frac{1}{4} \cdot 20 \sqrt{\pi} = 5 \sqrt{\pi}

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

\displaystyle A'= \left (5 \sqrt{\pi} \right )^{2}= 25 \pi

Example Question #11 : Calculating The Area Of A Square

The perimeter of a square is the same as the circumference of a circle with radius 8. What is the area of the square?

Possible Answers:

\displaystyle 16 \pi

\displaystyle 16 \pi^{2}

The correct answer is not among the other choices.

\displaystyle 4 \pi

\displaystyle 4 \pi^{2}

Correct answer:

\displaystyle 16 \pi^{2}

Explanation:

A circle with radius 8 has as its circumference \displaystyle 2 \pi times this, or 

\displaystyle 2 \pi \cdot 8 = 16 \pi.

This is also the perimeter of the square, so the sidelength is one fourth of this, or 

\displaystyle \frac{1}{4} \cdot 16 \pi = 4 \pi.

The area is the square of this, or

\displaystyle \left ( 4 \pi \right )^{2} = 16 \pi^{2}.

Example Question #15 : Calculating The Area Of A Square

The perimeter of a square is the same as the length of the hypotenuse of a right triangle with legs 8 and 12. What is the area of the square?

Possible Answers:

\displaystyle 13

\displaystyle 208

\displaystyle 52

\displaystyle 26

The correct answer is not among the other responses.

Correct answer:

\displaystyle 13

Explanation:

The length of the hypotenuse of a right triangle with legs 8 and 12 can be determined using the Pythagorean Theorem:

\displaystyle \sqrt{8^{2}+12^{2}} = \sqrt{64+144}= \sqrt{208} = \sqrt{16} \cdot \sqrt{13} = 4\sqrt{13}

Since this is also the perimeter of the square, its sidelength is one fourth of this, or

\displaystyle \frac{ 4\sqrt{13} }{4} =\sqrt{13}

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

\displaystyle \left ( \sqrt{13} \right )^{2}= 13

Example Question #11 : Calculating The Area Of A Square

If the perimeter of a square is \displaystyle 108, what is its area?

Possible Answers:

\displaystyle 10.4

\displaystyle 27

\displaystyle 729

\displaystyle 820

\displaystyle 1008

Correct answer:

\displaystyle 729

Explanation:

The perimeter of a square, and any shape for that matter, is found by adding up all the exterior sides. Since all sides are equal in a square, we can say: \displaystyle P=x + x + x + x=4x

where \displaystyle x represents the length of a side

We can solve for the side length using the information provided:

\displaystyle 4x=108

\displaystyle x=27

The area of a square is found by squaring the side length: 

\displaystyle A=x^2=27^2=729

Example Question #21 : Squares

The perimeter of a square is \displaystyle 4 ^{d}. Give its area.

Possible Answers:

\displaystyle 4 ^{d-2}

\displaystyle 4 ^{d+3}

\displaystyle 4 ^{d+1}

\displaystyle 4 ^{2d- 1}

\displaystyle 4 ^{2d- 2 }

Correct answer:

\displaystyle 4 ^{2d- 2 }

Explanation:

The length of one side of a square is the perimeter divided by 4:

\displaystyle \frac{4 ^{d} }{4} = \frac{4 ^{d} }{4 ^{1}} = 4 ^{d-1}

Square this to get the area:

\displaystyle ( 4 ^{d-1} )^{2} = 4 ^{(d-1) \cdot 2 } = 4 ^{2d- 2 }

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