# GMAT Math : Polygons

## Example Questions

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### Example Question #1 : Polygons

What is the perimeter of a hexagon?

1) Each side measures 10 cm

2) The hexagon is regular.

BOTH statements TOGETHER are sufficient, but neither statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements 1 and 2 TOGETHER are not sufficient.

Statement 2 ALONE is sufficient, but Statement 1 alone is not sufficient.

Statement 1 ALONE is sufficient, but Statement 2 alone is not sufficient.

Statement 1 ALONE is sufficient, but Statement 2 alone is not sufficient.

Explanation:

The perimeter is the sum of the measures of the sidelengths.

Knowing that the hexagon is regular only tells you the six sides are congruent; without the measure of any side, this does not help you.

Knowing only that each of the six sides measures 10 cm is by itself enough to calculate the perimeter to be

.

The answer is that Statement 1 is sufficient, but not Statement 2.

### Example Question #431 : Geometry

Note: Figure NOT drawn to scale

What is the perimeter of the above figure?

Assume all angles shown in the figure are right angles.

Not enough information is given to answer the question.

Explanation:

This figure can be seen as a smaller rectangle cut out of a larger one; refer to the diagram below.

We can fill in the missing sidelengths using the fact that a rectangle has congruent opposite sides. Once this is done, we can add the lengths of the sides to get the perimeter:

feet.

### Example Question #3 : Polygons

What is the perimeter of a rectangle with a length of  and a width of ?

Explanation:

The perimeter  of any figure is the sum of the lengths of its sides. Since we have a rectangle with a length of  and a width of , we know that there will be two sides of length  and two sides of width . Therefore:

### Example Question #4 : Polygons

What is the perimeter of a right triangle with a base of  and a height of ?

Not enough information provided

Explanation:

In order to find the perimeter  of the right triangle, we need to know the lengths of each of its sides. While we are given two sides - the base  and the height  - we do not know the hypotenuse . There are two ways that we can find , the first of which is the direct application of the Pythagorean Theorem:

We could have also noted that  is a common Pythagorean Triple and deduced the value of  that way.

Now that we have all three side lengths, we can calculate :

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

What is the perimeter of an octagon with equal side lengths of  each?

Explanation:

Starting with the knowledge that we are dealing with an octagon, an 8-sided figure, we calculate the perimeter  by adding the lengths of all 8 sides. Since we also know that each side measures , we can use multiplication:

### Example Question #1 : Polygons

is a pentagon with two sets of congruent sides and one side that is longer than all the others.

The smallest pair of congruent sides are 5 inches long each.

The other two congruent sides are 1.5 times bigger than the smallest sides.

The last side is twice the length of the smallest sides.

What is the perimeter of ?

Explanation:

A pentagon is a 5 sided shape. We are given that two sides are 5 inches each.

Side 1 = 5inches

Side 2 = 5 inches

The next two sides are each 1.5 times bigger than the smallest two sides.

Side 3 =Side 4= 7.5 inches

The last side is twice the size of the smallest side,

Side 5 =10 inches

Add them all up for our perimeter:

5+5+7.5+7.5+10=35 inches long

### Example Question #7 : Polygons

One side of a regular dodecagon has a length of .  What is the perimeter of the polygon?

Explanation:

A regular dodecagon is a polygon with twelve sides of equal length, so if one side has a length of , then the perimeter will be equal to twelve times the length of that one side. This gives us:

### Example Question #1 : Polygons

The hexagon in the above diagram is regular. If  has length 12, which of the following expressions is equal to the length of  ?

Explanation:

is a diameter of the regular hexagon. Examine the diagram below, which shows the hexagon with all three diameters:

Each interior angle of a hexagon measures , so, by symmetry, each base angle of the triangle formed is ; also, each central angle measures one sixth of , or . Each triangle is equilateral, so if , it follows that , and .

### Example Question #2 : Calculating The Length Of A Diagonal Of A Polygon

The octagon in the above diagram is regular. If  has length 8, which of the following expressions is equal to the length of  ?

Explanation:

Construct two other diagonals as shown.

Each of the interior angles of a regular octagon have measure , so it can be shown that  is a 45-45-90 triangle. Its hypotenuse is , whose length is 8, so, by the 45-45-90 Triangle Theorem, the length of  is 8 divided by :

Likewise, .

Since Quadrilateral  is a rectangle, .

### Example Question #1 : Calculating The Length Of A Diagonal Of A Polygon

Note: Figure NOT drawn to scale.

Which of the following statements is true of the length of  ?

The length of  is between 17 and 18.

The length of  is between 19 and 20.

The length of  is between 20 and 21.

The length of  is between 18 and 19.

The length of  is between 21 and 22.

The length of  is between 17 and 18.

Explanation:

By dividing the figure into rectangles and taking advantage of the fact that opposite sides of rectangles are congruent, we have the following sidelengths:

is the hypotenuse of a triangle with legs of lengths 8 and 16, so its length can be calculated using the Pythagorean Theorem:

The question can now be answered by noting that  and

,

so  falls between 17 and 18.

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