# GRE Math : Plane Geometry

## Example Questions

### Example Question #31 : Geometry

What is the perimeter of a pie piece if the pie is sliced into 40 degree pieces and its area is 361π?

Possible Answers:

40.1π

2.1

38π/9

38π

38 + 38π/9

Correct answer:

38 + 38π/9

Explanation:

The perimeter of a given pie-piece will be equal to 2 radii plus the outer arc (which is a percentage of the circumference). For our piece, this arc will be 40/360 or 1/9 of the circumference. If the area is 361π, this means πr2 = 361 and that the radius is 19. The total circumference is therefore 2 * 19 * π or 38π.

The total perimeter of the pie piece is therefore 2 * r + (1 / 9) * c = 2 * 19 + (1/9) * 38π = 38 + 38π/9

### Example Question #32 : Geometry

What is the length of the arc of a circle with radius 10 that traces a 50 degree angle?

Possible Answers:

25π

25π/29

20π/9

25π/7

25π/9

Correct answer:

25π/9

Explanation:

length of an arc = (degrees * 2πr)/360 = (50 * 2π * 10)/360 = 25π/9

### Example Question #33 : Geometry

An ant walks around the edge of circular pizzas left on the counter of a pizza shop. On most days, it is shaken off the pizza before it manages to walk the complete distance.

Quantity A: The distance covered by the ant when walking over four slices of a pizza with a diameter of  and  equally-sized pieces.

Quantity B: The distance covered by the ant when walking over a complete personal pizza with a diameter of  inches.

What can we say about the two quantities?

Possible Answers:

The relationship between the two quantities cannot be determined.

Quantity B is larger.

Quantity A is larger.

The two quantities are equal.

Correct answer:

Quantity B is larger.

Explanation:

Let's compute each quantity.

Quantity A

This is a little bit more difficult than quantity B.  It requires us to compute an arc length, for the ant does not walk around the whole pizza; however, this is not very hard. What we know is that the ant walks around , or , of the pizza.

Now, we know the circumference of a circle is  or

For our example, let's use the latter. Since , we know:

However, our ant walks around only part of this, namely:

Quantity B

This is really just a matter of computing the circumference of the circle.  For our value, we know this to be:

Now, we can compare these by taking quantity A and reducing the fraction to be . Thus, we know that Quantity B is larger than Quantity A.

### Example Question #31 : Circles

A circle of radius , for an angle , has an arc length of . What is the angle ?

Possible Answers:

Correct answer:

Explanation:

The formula for an arc length, , of a circle of a given radius, , and a given angle,  is:

Note that  is the circumference of the circle.

Conversely, for a known arc length and unknown angle, this equation can be rewritten as follows:

Plugging in the given values, it is therefore possible to find the missing angle:

### Example Question #2 : How To Find The Percentage Of A Sector From An Angle

Find the radius of the given circle:

Possible Answers:

The answer cannot be determined from the information given.

Correct answer:

Explanation:

To solve this problem, realize that an inscribed angle (an angle formed by two chords) is equal to twice the central angle formed by connecting the origin to the inscribed angle's endpoints:

Now, the formula for an arc length, , of a circle of a given radius, , and a given angle,  is:

Since the radius is the unknown, this equation can be rewritten as:

Plugging in our values, we find:

### Example Question #1 : Sectors

A given pizza with a -inch diameter has  calories. A baker cuts the pizza using a  angle for each piece. If Susan eats five such pieces, how many calories does she consume? Round to the nearest calorie.

Possible Answers:

Correct answer:

Explanation:

To solve this, notice that one piece of pizza comprises  of the total pie or (reducing)  of a pie. Now, if Susan buys five slices, she gets:

of the pie. If the complete pie contains  calories, she will then eat the following amount of calories:

Rounding, this is  calories.

### Example Question #1 : How To Find The Angle For A Percentage Of A Circle

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?

Possible Answers:

81.53°

91.44°

74.76°

128.21°

149.52°

Correct answer:

149.52°

Explanation:

To solve this, we must ascertain the following:

1) The arc length through which the ant traveled.

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

3) 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°.

### Example Question #1 : How To Find The Angle For A Percentage Of A Circle

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?

Possible Answers:

18°

60°

Insufficient information to answer this question

10°

36°

Correct answer:

18°

Explanation:

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°.

### Example Question #12 : Sectors

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?

Possible Answers:

Correct answer:

Explanation:

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 . This reduces to  degrees.

### Example Question #11 : Sectors

If the outer arc of 1/12th of a circular pie is 7π, what is the area of 1/4th of the pie?

Possible Answers:

1764π

441π

10.5π

None of the other answers

21π

Correct answer:

441π

Explanation:

Our initial data tells us that (1/12)c = 7π or (1/12)πd = 7π.  This simplifies to (1/12)d = 7 or d = 84. Furthermore, we know that r is 42. Given this, we can ascertain the area of a quarter of the whole pie by taking one fourth of the whole area or 0.25 * π * 422 = 0.25 * 1764 * π = 441π

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