GMAT Math : Calculating the area of a square

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Example Question #11 : Calculating The Area Of A Square

Squares

Note: Figure NOT drawn to scale

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

Possible Answers:

225:221

225:169

15:13

225:173

17:15

Correct answer:

225:173

Explanation:

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

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

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

The ratio is 225:173 .

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Example Question #171 : Geometry

Squares

Note: Figure NOT drawn to scale

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

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

Possible Answers:

$75 \%$

$90 \%$

$85 \%$

$70 \%$

$80 \%$

Correct answer:

$80 \%$

Explanation:

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

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

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

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

The area of Square WXYZ is

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

of that of Square ABCD .

Of the five choices, 80% comes closest.

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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:

$25 \pi$

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

$\frac{200 \sqrt{\pi}}{\pi}$

$50 \sqrt{\pi}$

$\frac{100 }{\pi}$

Correct answer:

$25 \pi$

Explanation:

The formula for the area of a circle is

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

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

$\pi r^{2} = 100$

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

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

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

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

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

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

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

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

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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:

$16 \pi$

$16 \pi^{2}$

The correct answer is not among the other choices.

$4 \pi$

$4 \pi^{2}$

Correct answer:

$16 \pi^{2}$

Explanation:

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

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

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

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

The area is the square of this, or

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

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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:

208

The correct answer is not among the other responses.

Correct answer:

Explanation:

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

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

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

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

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

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

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Example Question #11 : Calculating The Area Of A Square

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

Possible Answers:

10.4

729

820

1008

Correct answer:

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: P=x + x + x + x=4x

where represents the length of a side

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

4x=108

x=27

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

A=x^2=27^2=729

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Example Question #21 : Squares

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

Possible Answers:

$4 ^{d-2}$

$4 ^{d+3}$

$4 ^{d+1}$

$4 ^{2d- 1}$

$4 ^{2d- 2 }$

Correct answer:

$4 ^{2d- 2 }$

Explanation:

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

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

Square this to get the area:

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

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GMAT Math : Calculating the area of a square

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #11 : Calculating The Area Of A Square

Example Question #171 : Geometry

Example Question #61 : Quadrilaterals

Example Question #11 : Calculating The Area Of A Square

Example Question #15 : Calculating The Area Of A Square

Example Question #11 : Calculating The Area Of A Square

Example Question #21 : Squares

All GMAT Math Resources

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