# GRE Math : Squares

## Example Questions

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### Example Question #6 : 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?

Explanation:

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 #1 : 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

The information cannot be determined based on the information provided

Quantity B is greater

The two quantities are equal

Quantity A is greater

Quantity A is greater

Explanation:

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.

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

If a square has a side length of 4, how long is the diagonal of the square?

4

4√2

2√2

8√2

16

4√2

Explanation:

A diagonal divides a square into two 45-45-90 triangles, which have lengths adhering to the ratio of x: x: x√2. Therefore, 4√2 is the correct answer as the diagonal represents the hypotenuse of the triangle. the Pythagorean theorem can also be used: 42+ 42 = c2.

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

If a square has a side length of √2, how long is the diagonal of the square?

2√2

4-√2

4√2

4

2

2

Explanation:

A diagonal divides a square into two 45-45-90 triangles, which have lengths adhering to the ratio of x: x: x√2. Therefore, 2 is the correct answer as the diagonal represents the hypotenuse of the triangle. the Pythagorean theorem can also be used: √22+ √22 = c2.

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

A square has a side of length 5. What is the length of its diagonal?

5√3

5√2

5

10

10√3

5√2

Explanation:

The diagonal separates the square into two 45-45-90 right triangles. The problem can be solved by using the Pythagorean Theorem, a2 + b2 = c2.  It can also be solved by recognizing the 45-45-90 special triangles, which have side ratios of x√2.

### Example Question #181 : Geometry

A square with width of  is inscribed in a circle. What is the total area inside the circle?

Explanation:

We know that each side of the square is 6, so use the Pythagorean Theorem to solve for the diagonal of the square. The diagonal of the square is also the diameter of the circle.

Now let's find the area inside the circle using the radius.

meters

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

Quantity A:

The diagonal of a square with a side-length of .

Quantity B:

The side-length of a square with a diagonal of .

The relationship between the two quantities cannot be determined from the information given.

Quantity B is greater.

Quantity A is greater.

Both quantities are equal.

Both quantities are equal.

Explanation:

Quantity A: The diagonal of a square with a side-length of 7.

Quantity B: The side-length of a square with a diagonal of 14.

Both quantities can be determined, so can the relationship.

Quantity A:

To determine the diagonal of a square, it's important to remember that a diagonal directly bisects a square from corner to corner. In other words, it bisects the corners, creating two triangles with 45:45:90 proportions, with the diagonal serving as the hypotenuses . If you remember your special triangles, then the side-side-hypotenuse measurements have a ratio of

In Quantity A, the side-length is 7. Following the proportion:

The diagonal equals .

Quantity B:

We can use the same ratio to figure out quantity B by substituting x for the unknown side-length quantity, which looks like this:

To find x in this ratio, just isolate x in the hypotenuse:

Divide by

Now, how does Quantity A and Quantity B match up?

On the surface, it looks like the two quanties are equal. But how do we prove it?  Well, we know that .  Therefore, we know that:

Divide both sides by

Therefore, both quantities are equal.

### Example Question #181 : Plane Geometry

Quantity A:

The side-length of a square with a perimeter of .

Quantity B:

The side-length of a square with an area of .

Quantity B is greater

Quantity A is greater

Both quantities are equal

The relationship cannot be determined from the information given

Quantity A is greater

Explanation:

The first step to a quantitative comparison is to determine whether it can be solved at all with the given knowledge. Since all you need to find the side-length of a square is the perimeter, the area, OR the diagonal and we have one of each for these two quantities, this relationship can be determined. Thus, "the relationship cannot be determined" is out.

Now, to solve both quantities.  Quantity A can be solved by translating the perimeter into side lengths: the formula for the perimeter of a square is , with  being the side-length, so you just need to divide the perimeter by four.

Thus, quantity A is .

Quantity B can be solved by translating the area into side lengths: the formula for the area of a square is , or , with  being the side-length, so you just need to find the square root of the area.

Thus, quantity B is roughly .

Therefore quantity A is greater.

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

Circle  has a center in the center of Square .

If the area of Circle  is , what is the length of ?

Explanation:

If the area of Circle  is , we know that the area can be computed using the standard area formula:

, using  for

Simplifying, we get:

We know that  must be less than . By choosing  for , we find out that this is the radius of our circle. Thus, we know that the diameter of the circle is double this, or . Now, consider the following diagram:

Notice that the diameter is the same length as a side of the square. Thus,  is equal to .

### Example Question #3 : How To Find The Length Of The Side Of A Square

Circle  has a center in the center of square .

The line segment marked with length  lies on the diagonal of the square .

What is the length of side ?

This cannot be computed from the given data

Explanation:

You can further fill in your diagram as follows:

Now, we know that the triangle  is a  triangle. We also know that the length of  and  must also be equal to the diameter of the circle. (The diameter of the circle will run across the circle horizontally if you draw it that way. This will provide you with a complete side length.) Now, we know that the ratio of the hypotenuse of  to the side must be the same as:

For our data, that means:

Simplifying, we know:

Now, make both sides reciprocals:

Finally, solve:

Recall, this is both the length of the side and the diameter of the circle. Hence, you have your answer.

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