GMAT Math : Polygons

Study concepts, example questions & explanations for GMAT Math

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Example Questions

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Example Question #1 : Dsq: Calculating The Perimeter Of A Polygon

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.

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 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.

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.

Correct answer:

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 #1 : Dsq: Calculating The Perimeter Of A Polygon

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

Possible Answers:

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. 

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.

Correct answer:

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 : Dsq: Calculating The Perimeter Of A Polygon

Untitled

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

What is the perimeter of the above figure?

Statement 1: 

Statement 2: 

Possible Answers:

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

EITHER statement ALONE is 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.

Correct answer:

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.

Untitled

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 : Dsq: Calculating The Perimeter Of A Polygon

Consider regular decagon .

I) Side  is 56 inches long.

II) Side  plus Side  is equivalent to 112 inches.

Find the perimeter of .

Possible Answers:

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.

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

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

Correct answer:

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 : Dsq: Calculating The Perimeter Of A Polygon

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

Possible Answers:

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.

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

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

Either statement is sufficient to answer the question.

Correct answer:

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 #1 : Dsq: Calculating The Length Of A Diagonal Of A Polygon

Hexagon_44

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

Possible Answers:

Correct answer:

Explanation:

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

Untitled

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 #1 : Dsq: Calculating The Length Of A Diagonal Of A Polygon

Calculate the diagonal of a rectangle.

Statement 1:  The perimeter is 10.

Statement 2:  The area is 4.

 

Possible Answers:

Correct answer:

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 #131 : Geometry

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.

Possible Answers:

Either of the statements is sufficient.

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

Together the two statements are sufficient.

Statement 2 alone is sufficient.

Statement 1 alone is sufficient.

Correct answer:

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 #1 : Polygons

How many sides does a regular polygon have?

1) Each of the angles measures 140 degrees.

2) Each of the sides has measure 8.

Possible Answers:

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.

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.

BOTH statements TOGETHER are NOT sufficient to answer the question.

Correct answer:

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 #2 : Dsq: Calculating An Angle In A Polygon

Polygons_2

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.

Possible Answers:

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.

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. 

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

Correct answer:

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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