# GMAT Math : Squares

## Example Questions

### Example Question #2282 : Gmat Quantitative Reasoning

Calculate the length of the square.

Statement 1): The area is .

Statement 2): The diagonal is .

BOTH statements taken TOGETHER are sufficient to answer the question, but neither statement ALONE is sufficient.

BOTH statements TOGETHER are NOT sufficient, and additional data is needed to answer the question.

Statement 2) ALONE is sufficient, but Statement 1) ALONE is not sufficient to answer the question.

Statement 1) ALONE is sufficient, but Statement 2) ALONE is not sufficient to answer the question.

EACH statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Explanation:

Statement 1) gives the area of the square.  For all positive real numbers, the formula, , or , can be used to find either area or side length interchangeably.

Statement 2) mentions that the diagonal is 1, which is a positive real number.  The formula , can be used to also find the side length.

Either statement alone is sufficient to solve for the length of the square.

### Example Question #2283 : Gmat Quantitative Reasoning

Find the length of the quadrilateral.

Statement 1.)  The area of a quadrilateral is .

Statement 2.)  All interior angles of a quadrilateral are right angles.

BOTH statements taken TOGETHER are sufficient to answer the question, but neither statement ALONE is sufficient.

EACH statement ALONE is sufficient.

BOTH statements TOGETHER are NOT sufficient, and additional data is needed to answer the question.

Statement 2) ALONE is sufficient, but Statement 1) ALONE is not sufficient to answer the question.

Statement 1) ALONE is sufficient, but Statement 2) ALONE is not sufficient to answer the question.

BOTH statements TOGETHER are NOT sufficient, and additional data is needed to answer the question.

Explanation:

Statement 1) mentions that the area of a quadrilateral is 4.  This statement is insufficient to solve for the length of the square because the family of quadrilaterals include any 4-sided shape with 4 interior angles.  Examples of quadrilaterals are squares, rectangles, rhombus, and trapezoids, but the quadrilateral is not necessarily a square.

Statement 2) mentions that all four interior angles of a quadrilateral are right angles.  This narrows down the shape to either a square or a rectangle.  Both shapes have 4 right angles, but there is not enough information to determine if the shape is a square or a rectangle.

Therefore, neither statement is sufficient to solve for the length of a quadrilateral.

### Example Question #11 : Squares

Data sufficiency question- do not actually solve the question

Find the area of a square.

1. The length of one side of the square is 4.

2. The length of the diagonal of the square is 12.

Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

Each statement alone is sufficient

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient

Each statement alone is sufficient

Explanation:

Because all 4 sides of a square are equal, knowing the length of one side is sufficient to answer the question. Using the Pythagorean Theorem, you can calculate the length of 1 side of a square by knowing the length of a diagonal and then calculate the area.

### Example Question #12 : Squares

A circle is inscribed inside Square . The circle intersects the square at points . Give the area of the square.

Statement 1: The circle has area .

Statement 2: Arc  has length .

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

From Statement 1, the radius of the circle can be calculated by working backwards from the area formula, and the diameter can be calculated by doubling this.

From Statement 2, since, by the regularity of the square,  is one fourth of the circle, the length of  can be multiplied by four to get the circumference of the circle. This can be divided by  to obtain this diameter.

This diameter is equal to the the sidelength of the square in which it is inscribed, so it can be squared to obtain the area of the square. This makes each statement alone sufficient to answer the question.

### Example Question #3 : Dsq: Calculating The Area Of A Square

A circle is inscribed inside Square . The circle intersects the square at points . Give the area of the square.

Statement 1: The circle has circumference .

Statement 2:  is a  arc.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

The diameter of the inscribed circle is equal to the the sidelength of the square. From Statement 1, the circumference can be divided by  to obtain this measure, and this can be squared to obtain the area of the square.

Statement 2 gives extraneous information, as, by regularity of the figure, it is already known that  is one fourth of the circle and, subsequently, a  arc.

### Example Question #13 : Squares

The above figure shows a square inscribed inside a circle. What is the area of the black region?

Statement 1: The square has perimeter 40.

Statement 2: The circle has area .

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The area of the black region is one-fourth the difference of the areas of the circle and the square.

If  is the sidelength of the square, then the length of its diagonal - which is also the diameter of the circle - is, by the Pythagorean Theorem, , and the radius . Therefore, if you calculate either the radius or the sidelength, you can calculate the other, allowing you to find the areas of the circle and the square.

Statement 1 allows you to find the sidelength; just divide 40 by 4.

Statement 2 allows you to find the radius; just solve for  in the equation .

Therefore, either one gives you enough information to solve the problem.

### Example Question #5 : Dsq: Calculating The Area Of A Square

Find the area square R

I) The diagonal of square R is .

II) The perimeter of square R is .

Statement 1 is sufficient to solve the question, but statement 2 is not sufficient to solve the question.

Statement 2 is sufficient to solve the question, but statement 1 is not sufficient to solve the question.

Both statements taken together are sufficient to solve the question.

Neither statement is sufficient to solve the question. More information is needed.

Each statement alone is enough to solve the question.

Each statement alone is enough to solve the question.

Explanation:

You can use the diagonal of a square to find its side length via the ratio of a 45/45/90 triangle.

In this case our triangle will have side lengths of 15.

We can also divide our perimeter by 4 to get our side length, which is again 15.

Therefore, each statement alone is sufficient to solve the question.

### Example Question #6 : Dsq: Calculating The Area Of A Square

True or false:

Statement 1: A square with sidelength  has perimeter greater than 100.

Statement 2: A square with sidelength  has area greater than 40.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

Explanation:

Assume Statement 1 alone. The perimeter of a square is four times the length of a side, which here is ; the perimeter is greater than 100, so

Therefore, the question can be answered in the affirmative from Statement 1 alone.

Statement 2 alone gives insufficient information, however. The area of a square is the square of the length of a side. If the sidelength is , for example, the area is the square of this, which is 81; if the sidelength is , the area is 121. Both areas exceed 40, but only in the second case is .

### Example Question #2286 : Gmat Quantitative Reasoning

Gene is building a fence. He is using square fence posts and needs to know the total distance around one pole. Help him find the distance,

I) The fence will be  feet tall and  feet long.

II) The diagonal distance from one corner of a fence post to its other corner is   inches.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Explanation:

We are asked to find the perimeter of a square. To do that we need a side length.

I) Is irrelevant and is trying to distract you with other aspects of the question.

II) Gives you the diagonal of the square. The diagonal of a square creates two 45/45/90 triangles.

Use this knowledge to find the other side lengths and then the perimeter.

### Example Question #14 : Squares

Find the perimeter of the square.

1. The length of the diagonal of the square is .
2. The area of the square is .

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Each statement alone is sufficient to answer the question.

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Each statement alone is sufficient to answer the question.

Explanation:

Statement 1: The length of the diagonal of a square is found by multiplying the length of one side by the square root of . In this case, it is easy to see the length of the side is .

Knowing the length of the side, we can find the perimeter of the square:

Statement 2: We can use the given area to solve for the length of the side

where  represents the side's length

We can now find the perimeter in the same manner:

Therefore, each statement alone is sufficient to answer the question.

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