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Intermediate Geometry : Pentagons

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Example Question #8 : How To Find The Area Of A Pentagon

A regular pentagon has a side length of and an apothem length of 17.2 . Find the area of the pentagon.

Possible Answers:

900 square units

1,350 square units

215 square units

725 square units

1,075 square units

Correct answer:

1,075 square units

Explanation:

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 17.2 .

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

$A=\frac{base\times height}{2}$

$A=\frac{25\times 17.2}{2}=\frac{430}{2}=215$

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

$215\times 5= 1,075$

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Example Question #9 : How To Find The Area Of A Pentagon

A regular pentagon has a side length of and an apothem length of 4.8 . Find the area of the pentagon.

Possible Answers:

sq. units

16.8 sq. units

sq. units

133.6 sq. units

33.6 sq. units

Correct answer:

sq. units

Explanation:

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 4.8 .

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

$A=\frac{base\times height}{2}$

$A=\frac{7\times 4.8}{2}=\frac{33.6}{2}=16.8$

Note: 16.8 is only the measurement for one of the five interior triangles. Thus, the solution is:

$16.8\times5=84$

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

A regular pentagon has a perimeter of and an apothem length of 8.9 . Find the area of the pentagon.

Possible Answers:

57.85

280.50

275.85

289.25

Correct answer:

289.25

Explanation:

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

p=5(s)

65=5(s)

$s=\frac{65}{5}=13$

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 8.9 .

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

$A=\frac{base\times height}{2}$

$A=\frac{8.9\times 13}{2}=\frac{115.7}{2}=57.85$

Thus, the area of the entire pentagon is:

$A=57.85\times 5=289.25$

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Example Question #11 : How To Find The Area Of A Pentagon

Pentagon_series_vt_custom_

Find the area of the pentagon shown above.

Possible Answers:

184 sq. units

189 sq. units

sq. units

180 sq. units

198 sq. units

Correct answer:

198 sq. units

Explanation:

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:

$A=width\times length$

$A=18\times8=144$

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

$A=\frac{base\times height}{2}$

$A=\frac{9\times 6}{2}=\frac{54}{2}=27$

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:

A=144+27+27=198

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Example Question #12 : How To Find The Area Of A Pentagon

A regular pentagon has a perimeter of 145 and an apothem length of . Find the area of the pentagon.

Possible Answers:

450 sq. units

580 sq. units

1,580 sq. units

1,450 sq. units

1,500 sq. units

Correct answer:

1,450 sq. units

Explanation:

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

p=5(s)

145=5(s)

$s=\frac{145}{5}=29$

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:

$A=\frac{base\times height}{2}$

$A=\frac{29\times 20}{2}=\frac{580}{2}=290$

Thus, the area of the entire pentagon is:

$A=290\times 5=1,450$

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Example Question #13 : How To Find The Area Of A Pentagon

Find the area of the regular pentagon.

Possible Answers:

None of these

Correct answer:

Explanation:

Recall that the area of regular polygons can be found using the following formula:

$\text{Area of Regular Polygon}=\frac{1}{2}(\text{apothem})(\text{perimeter})$

First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:

$\text{Perimeter}=5(\text{side length})$

Substitute in the length of the given side of the pentagon in order to find the perimeter.

$\text{Perimeter}=5(3)$

$\text{Perimeter}=15$

Next, substitute in the given and calculated information to find the area of the pentagon.

$\text{Area of Pentagon}=\frac{1}{2}(2)(15)$

$\text{Area of Pentagon}=15$

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Example Question #14 : How To Find The Area Of A Pentagon

Find the area of the regular pentagon.

Possible Answers:

380

400

390

420

Correct answer:

420

Explanation:

Recall that the area of regular polygons can be found using the following formula:

$\text{Area of Regular Polygon}=\frac{1}{2}(\text{apothem})(\text{perimeter})$

First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:

$\text{Perimeter}=5(\text{side length})$

Substitute in the length of the given side of the pentagon in order to find the perimeter.

$\text{Perimeter}=5(14)$

$\text{Perimeter}=70$

Next, substitute in the given and calculated information to find the area of the pentagon.

$\text{Area of Pentagon}=\frac{1}{2}(12)(70)$

$\text{Area of Pentagon}=420$

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Example Question #15 : How To Find The Area Of A Pentagon

Find the area of the regular pentagon.

Possible Answers:

Correct answer:

Explanation:

Recall that the area of regular polygons can be found using the following formula:

$\text{Area of Regular Polygon}=\frac{1}{2}(\text{apothem})(\text{perimeter})$

First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:

$\text{Perimeter}=5(\text{side length})$

Substitute in the length of the given side of the pentagon in order to find the perimeter.

$\text{Perimeter}=5(8)$

$\text{Perimeter}=40$

Next, substitute in the given and calculated information to find the area of the pentagon.

$\text{Area of Pentagon}=\frac{1}{2}(5)(40)$

$\text{Area of Pentagon}=60$

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Example Question #16 : How To Find The Area Of A Pentagon

Find the area of the regular pentagon.

Possible Answers:

Correct answer:

Explanation:

Recall that the area of regular polygons can be found using the following formula:

$\text{Area of Regular Polygon}=\frac{1}{2}(\text{apothem})(\text{perimeter})$

First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:

$\text{Perimeter}=5(\text{side length})$

Substitute in the length of the given side of the pentagon in order to find the perimeter.

$\text{Perimeter}=5(7)$

$\text{Perimeter}=35$

Next, substitute in the given and calculated information to find the area of the pentagon.

$\text{Area of Pentagon}=\frac{1}{2}(4)(35)$

$\text{Area of Pentagon}=70$

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Example Question #17 : How To Find The Area Of A Pentagon

Find the area of the regular pentagon.

Possible Answers:

630

560

700

420

Correct answer:

560

Explanation:

Recall that the area of regular polygons can be found using the following formula:

$\text{Area of Regular Polygon}=\frac{1}{2}(\text{apothem})(\text{perimeter})$

First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:

$\text{Perimeter}=5(\text{side length})$

Substitute in the length of the given side of the pentagon in order to find the perimeter.

$\text{Perimeter}=5(16)$

$\text{Perimeter}=80$

Next, substitute in the given and calculated information to find the area of the pentagon.

$\text{Area of Pentagon}=\frac{1}{2}(14)(80)$

$\text{Area of Pentagon}=560$

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Intermediate Geometry : Pentagons

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #8 : How To Find The Area Of A Pentagon

Example Question #9 : How To Find The Area Of A Pentagon

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

Example Question #11 : How To Find The Area Of A Pentagon

Example Question #12 : How To Find The Area Of A Pentagon

Example Question #13 : How To Find The Area Of A Pentagon

Example Question #14 : How To Find The Area Of A Pentagon

Example Question #15 : How To Find The Area Of A Pentagon

Example Question #16 : How To Find The Area Of A Pentagon

Example Question #17 : How To Find The Area Of A Pentagon

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