# GRE Math : Plane Geometry

## Example Questions

### Example Question #21 : Geometry

A circular fence around a monument has a circumference of  feet. What is the radius of this fence?

Explanation:

This question is easy on the whole, though you must not be intimidated by one small fact that we will soon see. Set up your standard circumference equation:

The circumference is  feet, so we can say:

Solving for , we get:

Some students may be intimidated by having  in the denominator; however, there is no need for such intimidation. This is simply the answer!

### Example Question #22 : Geometry

Circle  has a center in the center of Square .

The area of Square ABCD is  .

What is the radius of Circle ?

Explanation:

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:

### Example Question #21 : Geometry

The formula to find the radius of the largest circle that can fit in an equilateral triangle is , where  is the length of any one side of the triange.

What is the largest diameter of a circle that can fit inside an equilateral triangle with a perimeter of  cm?

cm

cm

cm

cm

cm

Explanation:

The diameter is

To solve for the largest diameter multiply each side by 2.

The resulting formula for diamenter is

.

Substitute in 5  for S and solve. Diameter =  = 2.89 cm

### Example Question #24 : Geometry

Quantity A: The diameter of a circle with area of

Quantity B: The diameter of a circle with circumference of

Which of the following is true?

Quantity A is larger.

Both quantities are equal.

Quantity B is larger.

The relationship of the quantities cannot be determined.

Quantity B is larger.

Explanation:

Consider each quantity separately.

Quantity A

Recall that the area of a circle is defined as:

We know that the area is . Therefore,

Divide both sides by :

Therefore, . Since , we know:

Quantity B

This is very easy. Recall that:

Therefore, if . Therefore, Quantity B is larger.

### Example Question #25 : Geometry

Quantity A: The diameter of a circle with area of

Quantity B: The diameter of a circle with circumference of

Which of the following is true?

The two quantities are equal.

The relationship between the quantities cannot be determined.

Quantity B is larger.

Quantity A is larger.

Quantity B is larger.

Explanation:

Consider each quantity separately.

Quantity A

Recall that the area of a circle is defined as:

We know that the area is . Therefore,

Divide both sides by :

Therefore, .  Since , we know:

Quantity B

This is very easy.  Recall that:

Therefore, if .

Now, since your calculator will not have a square root button on it, we need to estimate for Quantity A. We know that  is . Therefore, .  This means that . Therefore, Quantity B is larger.

### Example Question #26 : Geometry

A circle with an area of  is divided into sectors with areas in a ratio of . What is the area of the largest sector?

Explanation:

From the ratio given , it may be easier to write it such that the terms sum up to . This can be taken by dividing each term by the sum of the terms:

or

The largest sector thus has an area equal to

### Example Question #27 : Geometry

A rectangle is inscribed inside of a circle such that every corner touches the edge of the circle. If the area of the rectangle is  and the perimeter of the rectangle is , what is the area of the circle in inches squared?

Explanation:

To find the area of the circle, it is important to know either its diameter or radius. For the geometry described in this problem, this is the same as the diagonal of the rectangle.

However, to find the diagonal of the rectangle, the sides must first be known. They can be found, since the perimeter and area are given:

This system of equation can be solved by substitution:

Followed by:

Note that this gives two possible values for ,  or , though the one selected is irrelevant; the other value will be the value for .

Knowing these two values, the diagonal can be found; it is the hypotenuse of a right triangle formed by these two lengths:

Since the diagonal is also the diameter, the area of the circle is given by:

### Example Question #21 : Geometry

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?"

The answer cannot be determined from the above information.

90 degrees

86.4 degrees

25.0 degrees

43.2 degrees

86.4 degrees

Explanation:

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.

### Example Question #29 : Geometry

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?

Explanation:

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

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:

Simplifying, we get:

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

### Example Question #30 : Geometry

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

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

Which of the following relations is true?

The two quantities are equal.

Quantity A is larger than quantity B.

The relationship between the two values cannot be determined.

Quantity B is larger than quantity A.

Quantity A is larger than quantity B.

Explanation:

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:

or, for our data:

Thus, our pertinent percentage is:

Quantity B

For this, the area is computed by the formula:

or, for our data:

Thus, our percentage is:

Clearly, , so A is larger than B.

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