# Intermediate Geometry : Pentagons

## Example Questions

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### Example Question #1 : How To Find An Angle In A Pentagon

The angles at 3 verticies of a pentagon are 60, 80 and 100.  Which of the following could NOT be the measures of the other 2 angles?

Explanation:

The sum of the angles in a polygon is

For a pentagon, this equals 540. Since the first 3 angles add up to 240, the remaining 2 angles must add up to

### Example Question #2 : How To Find An Angle In A Pentagon

What is the sum of three angles in a hexagon if the perimeter of the hexagon is ?

Explanation:

The perimeter in this question is irrelevant. Use the interior angle formula to determine the total sum of the angles in a hexagon.

There are six interior angles in a hexagon.

Each angle will be a sixth of the total angle.

Therefore, the sum of three angles in a hexagon is:

### Example Question #3 : How To Find An Angle In A Pentagon

Add four interior angles in a regular pentagon. What is the result?

Explanation:

Use the interior angle formula to find the total sum of angles in a pentagon.

for a pentagon, so substitute this value into the equation and solve:

Divide this number by 5, since there are five interior angles.

The sum of four interior angles in a regular pentagon is:

### Example Question #4 : How To Find An Angle In A Pentagon

What is the sum of two interior angles of a regular pentagon if the perimeter is 6?

Explanation:

The perimeter of a regular pentagon has no effect on the interior angles of the pentagon.

Use the following formula to solve for the sum of all interior angles in the pentagon.

Since there are 5 sides in a pentagon, substitute the side length .

Divide this by 5 to determine the value of each angle, and then multiply by 2 to determine the sum of 2 interior angles.

The sum of 2 interior angles of a pentagon is .

### Example Question #5 : How To Find An Angle In A Pentagon

Suppose an interior angle of a regular pentagon is .  What is ?

Explanation:

The pentagon has 5 sides. To find the value of the interior angle of a pentagon, use the following formula to find the sum of all interior angles.

Substitute .

Divide this number by 5 to determine the value of each interior angle.

Every interior angle is 108 degrees.  The problem states that an interior angle is .  Set these two values equal to each other and solve for .

### Example Question #6 : How To Find An Angle In A Pentagon

Let the area of a regular pentagon be .  What is the value of an interior angle?

Explanation:

Area has no effect on the value of the interior angles of a pentagon. To find the sum of all angles of a pentagon, use the following formula, where  is the number of sides:

There are 5 sides in a pentagon.

Divide this number by 5 to determine the value of each angle.

### Example Question #7 : How To Find An Angle In A Pentagon

True or false: Each of the five angles of a regular pentagon measures .

False

True

False

Explanation:

A regular polygon with  sides has  congruent angles, each of which measures

Setting , the common angle measure can be calculated to be

The statement is therefore false.

### Example Question #8 : How To Find An Angle In A Pentagon

True or false: Each of the exterior angles of a regular pentagon measures .

True

False

True

Explanation:

If one exterior angle is taken at each vertex of any polygon, and their measures are added, the sum is . Each exterior angle of a regular pentagon has the same measure, so if we let be that common measure, then

Solve for :

The statement is true.

### Example Question #1 : How To Find An Angle In A Pentagon

Given: Pentagon .

True, false, or undetermined: Pentagon  is regular.

False

Undetermined

True

Undetermined

Explanation:

Suppose Pentagon  is regular. Each angle of a regular polygon of  sides has measure

A pentagon has 5 sides, so set ; each angle of the regular hexagon has measure

Since one angle is given to be of measure , the pentagon might be regular - but without knowing more, it cannot be determined for certain. Therefore, the correct choice is "undetermined".

### Example Question #1 : How To Find The Perimeter Of A Pentagon

An apothem is a line drawn from the center of a regular shape to the center of one of its edges. The line drawn is perpendicular to the edge. The apothem of a regular pentagon is . What is the perimeter of this pentagon?

Explanation:

From the given information, we can imagine the following image

In order to solve for the perimeter, we need to solve for the length of one of the sides. This can be accomplished by creating a right triangle from the apothem and the top angle that's marked.

This angle measure can be calculated through:

, where the sum of all the angles around the center of the pentagon sum up to  and the  is for the number of sides. We can do this because all of the interior angles of the pentagon will be equal because it is regular. Then, this answer needs to be divided by  because the pictured right triangle is half of a larger isosceles triangle.

Therefore, the marked angle is

Now that one side and one angle are known, we can use SOH CAH TOA because this is a right triangle. In this case, the tangent function will be used because the mystery side of interest is opposite of the known angle, and we are already given the adjacent side (the apothem). In solving for the unknown side, we will label it as .

For a more precise answer, keep the entire number in your calculator. This will prevent rounding errors.

Keep in mind that the base of the triangle is actually only half the length of the side. This means that the value for  must be multiplied by

In order to solve for the perimeter, the length of the side needs to be multiplied by the number of sides; in this case, there are five sides.

This is the final answer, so the entire decimal is no longer needed. Rounded, the perimeter is .

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