DSQ: Calculating circumference - GMAT Quantitative Reasoning

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Question

Square S is inscribed in circle C as in the figure above. What is the circumference of C?

(1) The perimeter of S is 16.

(2) The area of S is 36.

Answer

From statement (1), we know that the side of S is 4, and then we can calculate the diagonal of S using the Pythagorean theorem: $(4^2 + 4^2)^\frac{1}{2} = 4\sqrt{2}$ . The diagonal of S is the diameter of C. Therefore, we can calculate the circumference by using $\pi d = 4\sqrt{2}\pi$ . From statement (2), we know that the side of S is 6, and then we can calculate the diagonal of S using the Pythagorean theorem:

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

The diagonal of S is the diameter of C. Therefore, we can calculate the circumference by using $\pi d= 6\sqrt{2}\pi$ .

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Question

What is the circumference of circle J?

I) Circle J has an area of $\small 64\pi in^2$ .

II) Circle J has a diameter of $\small16 in$ .

Answer

We are given the area and diameter of a circle and asked to find the circumference. We know that diameter is twice the length of a radius, so we also have our radius.

Given the following equations:

$\small a=\pi r^2$

$\small c=2\pi r$

We can see that knowing either diameter or area will allow us to find the circumference.

Thus: Each statement alone is enough to solve the question.

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Question

What is the circumference of the circle given by:

$\small (x-7)^2+(y-14)^2=r^2$

I) $\small r^2=225$ .

II) The slope of the tangent to the circle at is undefined.

Answer

All we need to find circumference is the radius.

I) Gives us the radius squared, so we could find circumference with I.

II) Tells us the slope of the tangent line at a given point is undefined. Only vertical lines have undefined slope. The tangent line is perpendicular to the radius, so we can find our radius by drawing a picture and comparing the location of the center to the location of the tangent line.

So either statement will be sufficient.

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Question

What is the circumference of Circle ?

1.) The diameter of the circle is .

2.) The area of the circle is $25\pi$ .

Answer

We are asked to find the circumference of Circle and are given the diameter and the area. We also know that $C=2\pi r$ . Taking each statement individually:

1.) The diameter is and we know that the radius $r=\frac{d}{2}$ , so $C=2\pi r=2\pi\left(\frac{d}{2}\right)=\pi d=10\pi$ . Therefore, Statement 1 is sufficient to solve for the circumference of the circle by itself.

2.) The area of Circle is $A=\pi r^{2}=25\pi$ , so we can determine that the radius . Since the circumference $C=2\pi r$ , Statement 2 is is sufficient to solve for the circumference of the circle by itself.

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Question

What is the circumference of Circle ?

1.) The radius of the circle is .

2.) The circle is inside another circle of area $81\pi$ .

Answer

We are asked to find the circumference of Circle and are given the diameter and the area. We also know that $C=2\pi r$ . Taking each statement individually:

1.) The radius is and we know that $C=2\pi r=2\pi(9)=18\pi$ . Therefore, Statement 1 is sufficient to solve for the circumference of the circle by itself.

2.) The area of the outside circle is $A=\pi r^{2}=81\pi$ , but we cannot use this to determine the circumference of Circle because we don't know where is inside the larger outside circle.

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