GMAT Quantitative - Calculating the length of the side of a square

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A circle is inscribed in a square. The area of the circle is $2\pi$ . What is the area of the square?

$2\sqrt{2}$

$3\sqrt{2}$

Explanation

Since we know the area of the circle, we can tell that: $\pi \cdot r^{2}= 2\pi$ . Where is the radius of the circle.

The length of a side of the square will be since the diameter of the circle is the same length as the side length of the square.

Finally we can calculate the area of the square which will be $(2r)^{2}$ . $r=\sqrt{2}$ so the area will be , which is our final answer.

Two squares in the same plane have the same center. The length of one side of the larger square is 10; the area of the region between the squares is 60. Give the length of one side of the smaller square.

$2 \sqrt{10}$

$4\sqrt{5}$

$6 \sqrt{2}$

Explanation

Let be the length of one side of the smaller square. Then the smaller square has area $S^{2}$ ; the larger square has area $10^{2} = 100$ . The area of the region between them, 60, is their difference:

$60 = 100 - S^{2}$

$60+ S^{2} - 60 = 100 - S^{2}+ S^{2} - 60$

$S^{2} = 40$

$S = \sqrt{40} = \sqrt{4} \cdot \sqrt{10} = 2 \sqrt{10}$

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A circle is inscribed in a square. The square has an area of , what is the perimeter of the circle?

$\sqrt{3}\pi$

$\sqrt{3}$

$3\pi$

$\pi$

$2\sqrt{3}\pi$

Explanation

The square as an area of 3. From that we can figure out the length of a side of the square, which is the same size as the diameter of the circle.

$A=s^2 \rightarrow 3=s^2$

$\sqrt3=s$

From the diameter of the circle we can find out, the perimeter of the circle given by the formula: $d\pi$ , where is the length of the diamter.

So $d=\sqrt{3}$ and therefore, the final answer is $\sqrt{3}\pi$ .

Two squares in the same plane have the same center. The length of one side of the smaller square is 10; the area of the region between the squares is 60. Give the length of one side of the larger square.

$4\sqrt{10}$

$8\sqrt{2}$

Explanation

Let be the length of one side of the larger square. Then the larger square has area $S^{2}$ ; the smaller square has area $10^{2} = 100$ . The area of the region between them, 60, is their difference:

$S^{2} - 100 = 60$

$S^{2} = 160$

$S = \sqrt{160} = \sqrt{16} \cdot \sqrt{10} = 4\sqrt{10}$

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is a square, and is the midpoint of diagonal . $BE=\sqrt{2}$ . What is the length of the side of the square?

$\frac{1}{2}$

$\frac{1}{3}$

$\sqrt{3}$

Explanation

Since BE is half of the diagonal BD, it follows that the size of the diagonal must be $2\sqrt{2}$ .

The length of a diagonal of a square will always be $s\sqrt{2}$ , where is the length of the side of the square.

Since the diagonal is $2\sqrt{2}$ , the side of the square must be 2.

The perimeter of a square plus its area is equal to . What is the length of a side of the square?

$\sqrt{7}$

$3\sqrt{2}$

Explanation

First we must set up the equation described by the first sentence of the problem. It tells us the perimeter of a square plus its area is equal to . We want to write this in terms of length, so we can write the following:

Where is the length of one side of the square. We then bring over the to write the equation as a polynomial that we can factor. This gives us:

A negative length makes no sense, so we can cross out the first result and we're left with the answer, . We could plug this length back into the original formula and see that a square with this side length would have a perimeter of and an area of , which add up to , verifying the solution.

Square A has twice the sidelength of Square B. The total area of the squares is 605 square feet. What is the sidelength of Square B?

$11\textrm{ ft}$

$13\textrm{ ft}$

$9\textrm{ ft}$

$12\textrm{ ft }6\textrm{ in}$

$10 \textrm{ ft } 6\textup{ in}$

Explanation

Let be the sidelength of Square B. Then the sidelength of Square A is , and the areas of Squares A and B are $\left ( 2s\right )^{2} = 4s$ and $s^{2}$ , respectively. Since the areas add up to 605 square feet, we solve for in this equation: