### All GRE Math Resources

## Example Questions

### Example Question #1 : How To Find The Area Of A Square

In the figure above, the circle is inscribed within the bounding square. If r = 5, what is the area of the shaded region?

**Possible Answers:**

100

25π

100π

100 - 25π

25 - (25/4)π

**Correct answer:**

25 - (25/4)π

In order to solve this, we must first find the area of the containing square and then remove the inscribed circle. Once this is done, we need to divide our result by 4 in order to get the one-forth that is the one shaded region.

One side of the square will be equal to the circle's diameter (2r). Since r = 5, d = 10. Therefore, the area of the square is d^{2} = 10^{2} = 100. The area of the circle is πr^{2} = 5^{2}π = 25π.

Therefore, the area of the four "corner regions" is equal to 100 - 25π. One of these is equal to (100 - 25π) / 4. Simplified, this is 25 - (25/4)π.

### Example Question #2 : How To Find The Area Of A Square

Find the area of a square with a side length of 4.

**Possible Answers:**

24

8

32

16

**Correct answer:**

16

All sides are equal in a square. To find the area of a square, multiply length times width. We know length = 4 but since all sides are equal, the width is also 4. 4 * 4 = 16.

### Example Question #2 : How To Find The Area Of A Square

If one doubles the radius of the semi-circle on the right of the diagram above, by what percentage does the overall area of the diagram change?

**Possible Answers:**

200%

100%

150%

300%

400%

**Correct answer:**

300%

To compare, first calculate the area of figure 1. Since it shares dimensions with the semi-circle, we will put all our variables in terms of the radius of that semi-circle:

A_{1} = (2r)^{2} + πr^{2}/2 = 4r^{2} + πr^{2}/2 = r^{2}(4 + π)/2

If we double r, we get:

A_{2} = (2 * 2r)^{2} + π(2r)^{2}/2 = 16r^{2} + π4r^{2}/2 = 4r^{2}(4 + π)/2

This means that the new figure is 4x the size of the original. This is an increase of 300%.

### Example Question #1 : How To Find The Area Of A Square

100 cm = 1 m

Quantity A: The area of a square with side 1 m

Quantity B: One hundred times the area of a square with side 1 cm

**Possible Answers:**

Quantity A is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Quantity B is greater.

**Correct answer:**

Quantity A is greater.

The obvious answer for this problem is that they are equal. Remember quantitative comparisons are often tricky and require you to check your initial inclination. If 1 m equals 100 cm then a square with side 1 m (100 cm) has an area of 100 cm x 100 cm or 10,000 cm^{2}. The area of a square with a 1 cm side is 1 cm x 1 cm or 1 cm^{2}. One hundred times 1 cm^{2 }is 100 cm^{2}. 10,000 cm^{2} is larger than 100 cm^{2} so Quantity A is greater.

### Example Question #2 : How To Find The Area Of A Square

A square is inscribed in a circle. The diameter of the circle is 10 feet. What is the area of the square?

**Possible Answers:**

It cannot be determined from the information provided.

**Correct answer:**

Since the diameter of the circle is 10, we know the radius of the circle is 5 feet. We can then draw radii that go from the center to two consecutive corners of the square. These radii are both 5 feet, and form a 90 degree angle (since they are diagonals of a square). Thus, with the enclosed side of the square they form a 45-45-90 triangle. Thus, the side of the square must be (this can also be found with the Pythagorean theorem). Then, the area of the square is

### Example Question #5 : How To Find The Area Of A Square

In the figure above, a square is inscribed in a circle with a diameter of 5 cm.

What is the area of the square?

**Possible Answers:**

**Correct answer:**

The diameter of the circle and the sides of the square form a 45-45-90 triangle. Since the hypotenuse is 5 cm, then a leg of the triangle (a side of the square) is .

The area of the square is then .

### Example Question #7 : How To Find The Area Of A Square

Square is on a coordinate plane, and each side of the square is parallel to either the -axis or -axis. Point has the coordinate and Point has the coordinate .

Quantity A: The area of square

Quantity B: 24

**Possible Answers:**

Quantity A is greater

The two quantities are equal

The information cannot be determined based on the information provided

Quantity B is greater

**Correct answer:**

Quantity A is greater

If you draw points and on the coordinate plane, you know that they are opposite ends of the square, since the sides are parallel to the axes. This means that the squares have sides with lengths of 5, making the area 25.

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