### All Intermediate Geometry Resources

## Example Questions

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

Find the area of the pentagon shown above.

**Possible Answers:**

square units

square units

square units

square units

square units

**Correct answer:**

square units

To find the area of this pentagon, divide the interior of the pentagon into a four-sided rectangle and two right triangles. The area of the bottom rectangle can be found using the formula:

The area of the two right triangles can be found using the formula:

Since there are two right triangles, the sum of both will equal the area of the entire triangular top portion of the pentagon.

Thus, the solution is:

### Example Question #2 : How To Find The Area Of A Pentagon

Find the area of the pentagon shown above.

**Possible Answers:**

square units

square units

square units

square units

**Correct answer:**

square units

To find the area of this pentagon, divide the interior of the pentagon into a four-sided rectangle and two right triangles. The area of the bottom rectangle can be found using the formula:

The area of the two right triangles can be found using the formula:

Since there are two right triangles, the sum of both will equal the area of the entire triangular top portion of the pentagon.

Thus, the solution is:

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

A regular pentagon has a side length of inches and an apothem length of inches. Find the area of the pentagon.

**Possible Answers:**

**Correct answer:**

By definition a regular pentagon must have equal sides and equivalent interior angles. Since we are told that this pentagon has a side length of inches, all of the sides must have a length of inches. Additionally, the question provides the length of the apothem of the pentagon--which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

Note: the area shown above is only the a measurement from one of the five total interior triangles. Thus, to find the total area of the pentagon multiply:

### Example Question #4 : How To Find The Area Of A Pentagon

A regular pentagon has a side length of and an apothem length of . Find the area of the pentagon.

**Possible Answers:**

square units

square units

square units

square units

square units

**Correct answer:**

square units

By definition a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon--which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

Note: is only the measurement for one of the five interior triangles. Thus, the final solution is:

### Example Question #5 : How To Find The Area Of A Pentagon

A regular pentagon has a perimeter of yards and an apothem length of yards. Find the area of the pentagon.

**Possible Answers:**

**Correct answer:**

To solve this problem, first work backwards using the perimeter formula for a regular pentagon:

Now you have enough information to find the area of this regular triangle.

Note: a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

Thus, the area of the entire pentagon is:

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

A regular pentagon has a side length of and an apothem length of . Find the area of the pentagon.

**Possible Answers:**

square units

square units

square units

square units

square units

**Correct answer:**

square units

By definition a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

Keep in mind that this is the area for only one of the five total interior triangles.

The total area of the pentagon is:

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

A regular pentagon has a perimeter of and an apothem length of . Find the area of the pentagon.

**Possible Answers:**

square units

square units

square units

square units

**Correct answer:**

square units

To solve this problem, first work backwards using the perimeter formula for a regular pentagon:

Now you have enough information to find the area of this regular triangle.

Note: a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

To find the total area of the pentagon multiply:

### Example Question #51 : Pentagons

A regular pentagon has a side length of and an apothem length of . Find the area of the pentagon.

**Possible Answers:**

square units

square units

square units

square units

square units

**Correct answer:**

square units

By definition a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

However, the total area of the pentagon is equal to:

### Example Question #52 : Pentagons

A regular pentagon has a side length of and an apothem length of . Find the area of the pentagon.

**Possible Answers:**

sq. units

sq. units

sq. units

sq. units

sq. units

**Correct answer:**

sq. units

By definition a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

Note: is only the measurement for one of the five interior triangles. Thus, the solution is:

### Example Question #53 : Pentagons

A regular pentagon has a perimeter of and an apothem length of . Find the area of the pentagon.

**Possible Answers:**

**Correct answer:**

To solve this problem, first work backwards using the perimeter formula for a regular pentagon:

Now you have enough information to find the area of this regular triangle.

Note: a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

Thus, the area of the entire pentagon is:

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