# GRE Math : Geometry

## Example Questions

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### Example Question #1 : Geometry

O is the center of the circle above.

The length of  is .

Quantity A: The area of the circle.

Quantity B:

Which of the following is true?

The two quantities are equal.

Quantity B is greater.

The relationship cannot be determined.

Quantity A is greater.

Quantity A is greater.

Explanation:

O is the center of the circle above.

The length of  is .

Quantity A: The area of the circle.

Quantity B:

Do not be tricked by this question. It is true that  can be split into halves, each of which are  in length. These halves are not, however, radii to the circle. Since this does not go through the center of the circle, its length is shorter than the diameter. This means that the radius of the circle must be greater than . Now, if it were , the area would be . Since it is larger than , the area must be larger than . Quantity A is larger than quantity B.

### Example Question #2 : Chords

O is the center of the circle above.

The circumference of the circle above is .

Quantity A: The length of .

Quantity B:

Which of the following is true?

Quantity A is larger.

The two quantities are equal.

Quantity B is larger.

The relationship cannot be determined.

Quantity B is larger.

Explanation:

Now, we know that the circumference of a circle is:

or

This means that the diameter of our circle is must be . Given this, we know that the  must be shorter than , for the diameter is the longer than any chord that does not pass through the center of the circle. Quantity B is larger than quantity A.

### Example Question #1 : Radius

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

32

15π

12π

12π

Explanation:

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

### Example Question #1 : Plane Geometry

Which is greater: the circumference of a circle with an area of , or the perimeter of a square with side length  inches?

The relationship cannot be determined from the information given.

The circumference of the circle is greater.

The two quantities are equal.

The perimeter of the square is greater.

The circumference of the circle is greater.

Explanation:

Starting with the circle, we need to find the radius in order to get the circumference. Find  by plugging our given area into the equation for the area of a circle:

Then calculate circumference:

(approximating  as 3.14)

To find the perimeter of the square, we can use  , where  is the perimeter and  is the side length:

, so the circle's circumference is greater.

### Example Question #1 : Circles

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

Explanation:

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.

### Example Question #4 : Geometry

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?

The relationship between the two values cannot be determined.

The two quantities are equal.

Quantity B is larger.

Quantity A is larger.

The relationship between the two values cannot be determined.

Explanation:

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 , we know:

Quantity B

If the diameter is one-fourth the radius of A, we know:

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

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

Therefore,

Now, notice that if , Quantity A is larger.

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

Quantity A:

Quantity B:

Therefore, the relation cannot be determined!

### Example Question #5 : Geometry

Circle  has a center in the center of Square .

The area of Square  is  .

What is the circumference 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:

The circumference of this circle is defined as:

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

### Example Question #1 : Geometry

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

π/3

121/π

225π

121π

121/π

Explanation:

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.

### Example Question #2 : Geometry

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

18π

18

324π

81π

81π

Explanation:

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.

### Example Question #1 : Plane Geometry

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?

4.5π

18π

3√(2)π

4.5π

Explanation:

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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