Circles - GRE Quantitative Reasoning

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Question

What is the circumference of a circle with an area of 36π?

Answer

We know that the area of a circle can be expressed: a = πr2

If we know that the area is 36π, we can substitute this into said equation and get: 36π = πr2

Solving for r, we get: 36 = r2; (after taking the square root of both sides:) 6 = r

Now, we know that the circuference of a circle is expressed: c = πd. Since we know that d = 2r (two radii, placed one after the other, make a diameter), we can rewrite the circumference equation to be: c = 2πr

Since we have r, we can rewrite this to be: c = 2π*6 = 12π

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Question

Circle A has an area of $121\pi$ . What is the perimeter of an enclosed semi-circle with half the radius of circle A?

Answer

Based on our information, we know that the 121π = πr2; 121 = r2; r = 11.

Our other circle with half the radius of A has a diameter equal to the radius of A. Therefore, the circumference of this circle is 11π. Half of this is 5.5π. However, since this is a semi circle, it is enclosed and looks like this:

Therefore, we have to include the diameter in the perimeter. Therefore, the total perimeter of the semi-circle is 5.5π + 11.

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Question

Which is greater: the circumference of a circle with an area of $25\pi \ in^2$ , or the perimeter of a square with side length inches?

Answer

Starting with the circle, we need to find the radius in order to get the circumference. Find $\dpi{100} \small r$ by plugging our given area into the equation for the area of a circle:

$A = \pi r^2$

$25\pi = \pi r^2$

$25 = r^2$

$r = 5\ in$

Then calculate circumference:

$C = 2\pi r$

$\dpi{100} \small C = 2\pi \times 5 = 10\pi \approx 31.4 inches$ (approximating $\dpi{100} \small \pi$ as 3.14)

To find the perimeter of the square, we can use $P = 4s$ , where $\dpi{100} \small P$ is the perimeter and $\dpi{100} \small s$ is the side length:

$P = 4 \times 7\ in = 28\ in$

$\dpi{100} \small 31.4>28$ , so the circle's circumference is greater.

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Question

$x\ge 1$

Quantity A: The circumference of a circle with radius

Quantity B: The area of a circle with a diameter one fourth the radius of the circle in Quantity A

Which of the following is true?

Answer

Let's compute each value separately. We know that the radii are positive numbers that are greater than or equal to . This means that we do not need to worry about the fact that the area could represent a square of a decimal value like .

Quantity A

Since $c=2\pi r$ , we know:

$a=2 \cdot \pi \cdot 24x = 48x\pi$

Quantity B

If the diameter is one-fourth the radius of A, we know:
$d = \frac{1}{4}\cdot 24x = 6x$

Thus, the radius must be half of that, or .

Now, we need to compute the area of this circle. We know:

$A = \pi r^2$

Therefore, $A = \pi \cdot (3x)^2 = 9x^2$

Now, notice that if , Quantity A is larger.

However, if we choose a value like , we have:

Quantity A: $48 * 1000 \pi = 48000\pi$

Quantity B: $9 * 1000^2\pi= 9000000\pi$

Therefore, the relation cannot be determined!

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Question

Circle has a center in the center of Square .

The area of Square is .

What is the circumference of Circle ?

Answer

Since we know that the area of Square is , we know , where is the length of one of its sides. From this, we can solve for by taking the square root of both sides. You will have to do this by estimating upward. Therefore, you know that is . By careful guessing, you can quickly see that is . From this, you know that the diameter of your circle must be half of , or (because it is circumscribed). Therefore, you can draw:

The circumference of this circle is defined as:

$C=2\pi r$ or, for your values:

$c=2\pi * 12 = 24\pi$

(You could also compute this from the diameter, but many students just memorize the formula above.)

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Question

An ant begins at the center of a pie with a 12" radius. Walking out to the edge of pie, it then proceeds along the outer edge for a certain distance. At a certain point, it turns back toward the center of the pie and returns to the center point. Its whole trek was 55.3 inches. What is the approximate size of the angle through which it traveled?

Answer

To solve this, we must ascertain the following:

The arc length through which the ant traveled.

The percentage of the total circumference in light of that arc length.

The percentage of 360° proportionate to that arc percentage.

To begin, let's note that the ant travelled 12 + 12 + x inches, where x is the outer arc distance. (It traveled the radius twice, remember); therefore, we know that 24 + x = 55.3 or x = 31.3.

Now, the total circumference of the circle is 2πr or 24π. The arc is 31.3/24π percent of the total circumference; therefore, the percentage of the angle is 360 * 31.3/24π. Since the answers are approximations, use 3.14 for π. This would be 149.52°.

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Question

A study was conducted to determine the effectiveness of a vaccine for the common cold (Rhinovirus sp.). 1000 patients were studied. Of those, 500 received the vaccine and 500 did not. The patients were then exposed to the Rhinovirus and the results were tabulated.

Table 1 shows the number of vaccinated and unvaccinated patients in each age group who caught the cold.

Suppose the scientists wish to create a pie chart reflecting a patient's odds of catching the virus depending on vaccination status and age group.

All 1000 patients are included in this pie chart.

What would be the angle of the arc for the portion of the chart representing vaccinated patients of all age groups who caught the virus?

Answer

First, we must determine what proportion of the 1000 patients were vaccinated and caught the virus. The total number of patients who were vaccinated and caught the virus is 50.

18 + 4 + 5 + 4 + 19 = 50

The proportion of the patients is represented by dividing this group by the total number of participants in the study.

50/1000 = 0.05

Next, we need to figure out how that proportion translates into a proportion of a pie chart. There are 360° in a pie chart. Multiply 360° by our proportion to reach the solution.

360° * 0.05 = 18°

The angle of the arc representing vaccinated patients who caught the virus is 18°.

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Question

A group of students ate an -inch pizza that was cut into equal slices. What was the angle measure needed to cut this pizza into these equal slices?

Answer

You will not need all of the information given in the prompt in order to answer this question successfully. You really only need to know that there were slices. If the slices were evenly divided among the degrees of the pizza, this means that the degree measure of each slice was $\frac{360}{14}$ . This reduces to $\frac{180}{7}$ degrees.

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Question

John owns 8 black shirts, 7 red shirts, 6 blue shirts and 4 white shirts. If he wants to make a circle chart of his shirts, what is the degree angle corresponding to the "blue shirt section?"

Answer

We can set up a ratio to calculate the angle measure as such: 6/25 = x/360, since there are 360 degrees in a circle. Solving, we obtain x = 86.4 degrees.

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Question

A circular pie is cut into 30 pieces. Two people wish to split a piece of the pie, but one person wants to have twice as much as the other person. What is the angle of the smaller piece produced in this manner?

Answer

First of all, calculate the angle of each of the full pieces of pie. This is easily found:

$\frac{360^{\circ}}{30}=12^{\circ}$

Though small, this is what the information tells us! Now, we know that if two people are eating the piece, with one having twice the amount of the other, the angles must be for the smaller piece and for the larger one. Thus, we can write the equation:

$x + 2x = 12^{\circ}$

Simplifying, we get:

$3x = 12^{\circ}$

$x = 4^{\circ}$

That is a tiny piece, but that is what is called for by the data!

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Question

Quantity A: The angle of a circle's sector having an arc length of $10\pi$ and a radius of .

Quantity B: The angle of a circle's sector having an area of $24\pi$ and a radius of .

Which of the following relations is true?

Answer

For each of these, you need to compute the total measurement applicable to the given data. For Quantity A, this will be the total circumference. For Quantity B, this will be the total area. You will then divide the given sector calculation by this total amount. By multiplying this percentage by , you will find the degree measure of each; however, you will merely need to stop at the percentage (since both are percentages of the same number, namely ).

Quantity A

The total circumference is calculated using the standard equation:

$C = 2\pi r$ or, for our data: $C = 2\pi * 7 = 14\pi$

Thus, our pertinent percentage is:

$\frac{10\pi}{14\pi} = \frac{5}{7}$

Quantity B

For this, the area is computed by the formula:

$A = \pi r^2$ or, for our data: $A = \pi*9^2 = 81\pi$

Thus, our percentage is:

$\frac{24\pi}{81\pi} = \frac{8}{27}$

Clearly, $\frac{5}{7} > \frac{8}{27}$ , so A is larger than B.

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Question

What is the angle between the hour and minute hand of a clock at 4:15?

Answer

In this case, the minute hand is pointing at 3 while the hour hand is pointing at some value past 4. Now, since 15 minutes represents 1/4 of an hour, the hour hand will be 1/4 the distance between 4 and 5. The total number of degrees between minutes is 360/12 = 30°; therefore, the hour hand is 30/4 or 7.5° past the 4 o'clock point. Now, between 3 and 4, there are 30°. Add to that the 7.5°. The total distance is therefore 37.5°.

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Question

What is the angle, in degrees, between the minute and hour hands of the clock at pm?

Answer

To find the degrees between hands on a clock, you must first remember that a clockface is a circle, and therefore has an internal angle sum of 360 degrees. Since a clockface has a sum of 360 degrees and there are 12 numbers on a clockface, the degree difference between each number is 360 divided by 12, or 30 degrees.

You can use this 30 degrees to figure out the distance between the minute and hour hands at the time given. At 5:40pm, the minute hand will be on the 8, as each number indicates five minutes on the clock.

It is the hour hand where this calculation gets tricky. While it would be easy to assume that the hour hand would be on the 5 at this time, that's actually incorrect. The hour hand slowly creeps towards the next number throughout the hour, moving in appropriate increments to reflect the fraction of the hour that has past. Since 40 minutes is 2/3 of an hour, then the hour hand is 2/3 of the way towards the number 6 on 5:40pm.

Thus, when calculating the distance, you must find the difference between 5 2/3 and 8 on the clock.

$8 -5\frac{2}{3}=2\frac{1}{3}$

Multiply this number by 30 (the degrees between each number on the clock, and thus the way we must convert this number to degrees), and you get 70 degrees.

$2\frac{1}{3}\cdot 30=\frac{7\cdot 30}{3}=\frac{210}{3}=70^\circ$

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Question

What is the area of a circle, one-quarter of the circumference of which is 5.5 inches?

Answer

Here, you need to “solve backward” from the data you have been given. We know that 0.25C = 5.5; therefore, C = 22. In order to solve for the area, we will need the radius of the circle. This can be obtained by recalling that C = 2πr. Replacing 22 for C, we get 22 = 2πr.

Solve for r: r = 22 / 2π = 11 / π.

Now, we solve for the area: A = πr2. Replacing 11 / π for r: A = π (11 / π)2 = (121π) / (π2) = 121 / π.

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Question

Given circle O with a diameter of 2 and square ABCD inscribed within circle O, what is the area of the shaded region?

Answer

There are two steps to this problem: determining the area of the circle and determining the area of the square. The area of the circle is πr2 which is π(2/1)2 or π. AD is a diameter of circle O and creates two isosceles right triangles with ACD and ABD. The relationship between sides of an isosceles right triangle is 1 : 1 : √2. Thus the sides of square ABCD are √2 and the area is 2. The area of the shaded region is the area of the circle minus the area of the square, or π – 2.

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Question

For , Chelsea can get either a $16\ in$ diameter pizza or two $8\ in$ diameter pizzas. Which is the better deal?

Answer

$A=\pi r^2$

$A_{16\ in}=8^2\pi =64\pi, \ A_{8\ in}=4^2\pi =16\pi$

$2(A_{8\ in})=2(16\pi )=32\pi$

Therefore the 16 inch pizza is the better deal.

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Question

Circle B has a circumference of 36π. What is the area of circle A, which has a radius half the length of the radius of circle B?

Answer

To find the radius of circle B, use the circumference formula (c = πd = 2πr):

2πr = 36π

Divide each side by 2π: r = 18

Now, if circle A has a radius half the length of that of B, A's radius is 18 / 2 = 9.

The area of a circle is πr2. Therefore, for A, it is π*92 = 81π.

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Question

In the diagram above, square ABCD is inscribed in the circle. If the area of the square is 9, what is the area of the circle?

Answer

If the area of the square is 9, then s2 = 9 and s = 3. If the sides thus equal 3, we can calculate the diagonals (either CB or AD) by using the 45-45-90 triangle ratio. For a side of 3, the diagonal will be 3√(2). Note that since the square is inscribed in the circle, this diagonal is also the diameter of the circle. If it is such, the radius is one half of that or 1.5√(2).

Based on that value, we can computer the circle’s area:

A = πr2 = π(1.5√(2))2 = (2.25 * 2)π = 4.5π

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Question

A small circle with radius 5 lies inside a larger circle with radius x. What is the area of the region inside the larger circle, but outside of the smaller circle, in terms of x?

Answer

Since the answers are in terms of pi, simply find the area of each circle in terms of x and ∏:

Smaller: ∏(5)2 = 25∏

Larger: ∏x2

We must subtract the inner circle from the outer circle; this translates to ∏x2-25∏.

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Question

Quantitative Comparison

Quantity A: Area of a circle with radius r

Quantity B: Perimeter of a circle with radius r

Answer

Try different values for the radius to see if a pattern emerges. The formulas needed are Area = π r_2 and Perimeter = 2_πr.

If r = 1, then the Area = π and the Perimeter = 2_π_, so the perimeter is larger.

If r = 4, then the area = 16_π_ and the perimeter = 8_π_, so the area is larger.

Therefore the relationship cannot be determined from the information given.

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