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Intermediate Geometry : How to find the area of a pentagon

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Example Questions

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

Pentagon_series_vt_custom_

Find the area of the pentagon shown above.

Possible Answers:

380 square units

340 square units

300 square units

320 square units

390 square units

Correct answer:

380 square 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:

$A=width\times length$

$A=20\times15=300$

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

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

$A=\frac{10\times 8}{2}=\frac{80}{2}=40$

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=300+40+40=380

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Example Question #781 : Intermediate Geometry

Pentagon_series_vt_custom_

Find the area of the pentagon shown above.

Possible Answers:

416 square units

502 square units

672 square units

572 square units

Correct answer:

572 square 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:

$A=width\times length$

$A=26\times16=416$

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

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

$A=\frac{13\times 12}{2}=\frac{156}{2}=78$

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=416+78+78=572

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Example Question #782 : Intermediate Geometry

A regular pentagon has a side length of inches and an apothem length of 6.2 inches. Find the area of the pentagon.

Possible Answers:

118 inches^2

inches^2

139.5 inches^2

inches^2

Correct answer:

139.5 inches^2

Explanation:

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 6.2 .

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

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

$A=\frac{9\times 6.2}{2}=\frac{55.8}{2}=27.9$

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:

$27.9\times5=139.5$

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Example Question #783 : Intermediate Geometry

A regular pentagon has a side length of and an apothem length of 9.6 . Find the area of the pentagon.

Possible Answers:

square units

200 square units

336 square units

67.2 square units

Correct answer:

336 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 9.6 .

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

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

$A=\frac{14\times 9.6}{2}=\frac{134.4}{2}=67.2$

Note: 67.2 is only the measurement for one of the five interior triangles. Thus, the final solution is:

$67.2\times5=336$

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Example Question #784 : Intermediate Geometry

A regular pentagon has a perimeter of yards and an apothem length of 8.3 yards. Find the area of the pentagon.

Possible Answers:

249 yards^2

yards^2

200 yards^2

yards^2

248 yards^2

Correct answer:

249 yards^2

Explanation:

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

p=5(s)

60=5(s)

$s=\frac{60}{5}=12$

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 8.3 .

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

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

$A=\frac{8.3\times 12}{2}=\frac{99.6}{2}=49.8$

Thus, the area of the entire pentagon is:

$A=49.8\times 5=249$

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Example Question #785 : Intermediate Geometry

A regular pentagon has a side length of and an apothem length of 10.3 . Find the area of the pentagon.

Possible Answers:

306.25 square units

400.5 square units

383.5 square units

386.25 square units

270 square units

Correct answer:

386.25 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 10.3 .

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

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

$A=\frac{15\times 10.3}{2}=\frac{154.5}{2}=77.25$

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

The total area of the pentagon is:

$A=77.25\times 5=386.25$

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Example Question #786 : Intermediate Geometry

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

Possible Answers:

175 square units

135 square units

square units

170 square units

Correct answer:

175 square units

Explanation:

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

p=5(s)

50=5(s)

$s=\frac{50}{5}=10$

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:

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

$A=\frac{10\times 7}{2}=\frac{70}{2}=35$

To find the total area of the pentagon multiply:

$35\times5=175$

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Example Question #2 : 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

1,075 square units

215 square units

725 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 #51 : Pentagons

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

Possible Answers:

133.6 sq. units

sq. units

33.6 sq. units

sq. units

16.8 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 #4 : 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:

280.50

57.85

289.25

275.85

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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Intermediate Geometry : How to find the area of a pentagon

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

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

Example Question #781 : Intermediate Geometry

Example Question #782 : Intermediate Geometry

Example Question #783 : Intermediate Geometry

Example Question #784 : Intermediate Geometry

Example Question #785 : Intermediate Geometry

Example Question #786 : Intermediate Geometry

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

Example Question #51 : Pentagons

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

All Intermediate Geometry Resources

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