# GMAT Math : Polygons

## Example Questions

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### Example Question #1 : Polygons

Given a regular hexagon and a regular octagon, which, if either, has the greater perimeter?

Statement 1: The sidelength of the octagon is one foot.

Statement 2: The sidelength of the hexagon is fifteen inches.

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

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.

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

Explanation:

Each of the two statements allows you to find out the perimeter of one of the polygons by multiplying its sidelength by the number of its sides. However, neither statement offers any clues to the perimeter of the other polygon. Both statements together, however, allow you to determine and to compare both perimeters.

### Example Question #2 : Polygons

You are given a regular hexagon and a regular pentagon. Which one has the greater perimeter?

Statement 1: The hexagon and the pentagon have the same area

Statement 2: The apothem of the hexagon is greater than that of the pentagon

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.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

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

Explanation:

The area of a polygon is one-half the product of its perimeter and its apothem (the perpendicular distance from the center to a side). Therefore,

or

Therefore, the perimeter can be determined from both area and apothem, but not from one alone. Neither statement alone gives you enough information, but from both statements together, it can be determined that the pentagon has the greater perimeter.

### Example Question #3 : Polygons

Note: Figure NOT drawn to scale. All angles shown are right angles.

What is the perimeter of the above figure?

Statement 1:

Statement 2:

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

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.

BOTH statements TOGETHER are insufficient to answer the question.

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

Explanation:

The figure can be seen as a  rectangle cut out of an  rectangle.

The perimeter of the composite figure is

.

However, since opposite sides of a rectangle are congruent, then, as can be seen in the figure,  and .

The perimeter can then be rewritten:

Therefore, it is necessary and sufficient to know  and ; the other four sidelengths are not needed to determine the perimeter of the figure.

### Example Question #4 : Polygons

Consider regular decagon .

I) Side  is 56 inches long.

II) Side  plus Side  is equivalent to 112 inches.

Find the perimeter of .

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.

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

Either statement is sufficient to answer the question.

Either statement is sufficient to answer the question.

Explanation:

To find perimeter, we need the length of all the sides. Note that we are dealing with a regular polygon, so all of its sides are equal.

Statement I gives us one side length. Multiply the side by ten (decagons have ten sides):

Statement II gives us the length of two sides together. Multiply by 5 to get the total perimeter:

### Example Question #5 : Polygons

Consider pentagon .

I) Side  has the same length as side , 5 inches.

II) Side  is one-third the length of side , and sides  and  have the same length as side .

What is the perimeter of

Both statements are needed to answer the question.

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

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

Either statement is sufficient to answer the question.

Both statements are needed to answer the question.

Explanation:

Consider pentagon .

I) Side  has the same length as side , 5 inches.

II) Side  is one-third the length of sides  and .

What is the perimeter of

To find perimeter, we need to know the length of all the sides.

Statement I gives us the lengths of two sides.

Statement II relates one of the sides given in Statement I to the other three sides:

### Example Question #6 : Polygons

The hexagon in the above diagram is regular. If  has length 10, which of the following expressions is equal to the length of  ?

Explanation:

The answer can be seen more easily by constructing the altitude of  from , as seen below:

Each interior angle of a hexagon measures ,and the altitude also bisects , the vertex angle of isosceles  is easily proved to be a 30-60-90 triangle, so by the 30-60-90 Triangle Theorem,

The altitude also bisects  at , so

.

### Example Question #7 : Polygons

Calculate the diagonal of a rectangle.

Statement 1:  The perimeter is 10.

Statement 2:  The area is 4.

Explanation:

Statement 1:  The perimeter is 10.

Given the perimeter of a rectangle is 10, set up the equation such that the sum of twice of length and width are equal to 10.

There are multiple combinations of length and width that will work in this scenario.

Statement 2:  The area is 4.

Write the formula for area of a rectangle and substitute the area.

Statements 1 and 2 require each other in order to solve for the length and width since there are two unknown variables.

Afterward, the Pythagorean Theorem can be used to solve for the diagonal of the rectangle.

Therefore:

### Example Question #8 : Polygons

A regular pentagon has been drawn on the side of a building by some mathematically minded graffiti artists. What is the length of a diagonal across it?

1) The length of a side is .

2) Each of the internal angles is  degrees.

Statement 1 alone is sufficient.

Together the two statements are sufficient.

Neither of the statements, separate or together, is sufficient.

Statement 2 alone is sufficient.

Either of the statements is sufficient.

Statement 1 alone is sufficient.

Explanation:

If the pentagon is regular, then it is known that each of the internal angles is  degrees, since the sum of the five interior angles of a pentagon is  degrees. Statement 2 does not give new information, nor does it give enough.

Statement 1 does, however. The length of the diagonal can be found by using the law of cosines:

and c is the length of the diagonal.

### Example Question #251 : Data Sufficiency Questions

How many sides does a regular polygon have?

1) Each of the angles measures 140 degrees.

2) Each of the sides has measure 8.

BOTH statements TOGETHER are 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 1 or Statement 2 ALONE is sufficient to answer the question.

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

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

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

Explanation:

The relationship between the number of sides of a regular polygon  and the measure of a single angle  is

If we are given that , then we can substitute and solve for   :

making the figure a nine-sided polygon.

Knowing only the measure of each side is neither necessary nor helpful; for example, it is possible to construct an equilateral triangle with sidelength 8 or a square with sidelength 8.

The correct answer is that Statement 1 alone is sufficient to answer the question, but not Statement 2 alone.

### Example Question #10 : Polygons

Note: Figure NOT drawn to scale.

Is the pentagon in the diagram above a regular pentagon?

Statement 1:

Statement 2: The hexagon in the diagram is not a regular hexagon.

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:

Information about the hexagon is irrelevant, so Statement 2 has no bearing on the answer to the question.

The measures of the exterior angles of any pentagon, one per vertex, total , and they are congruent if the pentagon is regular, so if this is the case, each would measure . But if Statement 1 is true, then an exterior angle of the pentagon measures  . Therefore, Statement 1 is enough to answer the question in the negative.

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