# ISEE Upper Level Math : Other Polygons

## Example Questions

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### Example Question #1 : Other Polygons

If the perimeter of a septagon is equal to , what is the length of one side? (All sides of the septagon are equal.)

Explanation:

If the perimeter of a septagon (in which all sides are equal) is , then the length of one side will be one seventh of this expression.

To find one seventh, the value must first be simplified and then divided by 7.

When this is divided by 7, the result is:

This value is thus the length of one side of the septagon.

### Example Question #1 : How To Find An Angle In Other Polygons

How many degrees are in an internal angle of a regular heptagon?

Explanation:

The number of degrees in an internal angle of a regular polygon can be solved using the following equation where n equals the number of sides in the polygon:

### Example Question #1 : How To Find An Angle In Other Polygons

What is the measure of an interior angle of a regular nonagon?

Explanation:

The measure of an interior angle of a regular polygon can be determined using the following equation where n equals the number of sides:

### Example Question #1 : Other Polygons

Note: Figure NOT drawn to scale.

Refer to the above diagram. Pentagon  is regular. What is the measure of  ?

Explanation:

The answer can be more clearly seen by extending  to a ray :

Note that angles have been newly numbered.

and  are exterior angles of a (five-sided) regular pentagon in relation to two parallel lines, so each has a measure of  is a corresponding angle to , so its measure is also .

### Example Question #1 : How To Find An Angle In Other Polygons

In the above figure, the seven-side polygon, or heptagon, shown is regular. What is the measure of  ?

The correct answer is not given among the other responses.

Explanation:

The answer can be more clearly obtained by extending the top of the two parallel lines as follows:

Note that two angles have been newly labeled.

is an interior angle of a regular heptagon and therefore has measure

By the Isosceles Triangle Theorem, since the two sides of the heptagon that help form the triangle are congruent, so are the two acute angles, and

is supplementary to , so

### Example Question #11 : Isee Upper Level (Grades 9 12) Mathematics Achievement

In the above figure, the seven-side polygon, or heptagon, shown is regular. What is the measure of  ?

The correct answer is not given among the other responses.

Explanation:

The answer can be more clearly seen by extending the lower right side of the heptagon to a ray, as shown:

Note that angles have been newly numbered.

and  are exterior angles of a (seven-sided) regular heptagon, so each has a measure of  is a corresponding angle to  in relation to two parallel lines, so its measure is also .

### Example Question #19 : Plane Geometry

Note: Figure NOT drawn to scale.

In the above figure, Pentagon  is regular. Give the measure of .

Explanation:

The sum of the degree measures of the angles of Quadrilateral  is 360, so

Each interior angle of a regular pentagon measures

,

which is therefore the measure of .

It is also given that  and , so substitute and solve:

### Example Question #20 : Plane Geometry

Note: Figure NOT drawn to scale.

In the above figure, Pentagon  is regular. Give the measure of .

The correct answer is not given among the other responses.

Explanation:

The sum of the degree measures of the angles of Quadrilateral  is 360, so

.

Each interior angle of a regular pentagon measures

,

which is therefore the measure of both  and .

and  form a linear pair, making them supplementary. Since ,

.

Substitute and solve:

### Example Question #21 : Isee Upper Level (Grades 9 12) Mathematics Achievement

The measures of the angles of an octagon form an arithmetic sequence. The greatest of the eight degree measures is . What is the least of the eight degree measures?

This octagon cannot exist.

Explanation:

The total of the degree measures of any eight-sided polygon is

.

In an arithmetic sequence, the terms are separated by a common difference, which we will call . Since the greatest of the degree measures is , the measures of the angles are

Their sum is

The least of the angle measures is

The correct choice is .

### Example Question #21 : Isee Upper Level (Grades 9 12) Mathematics Achievement

What is  of the total number of degrees in a 9-sided polygon?