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## Area of Pentagon

The area of a pentagon is the region that is enclosed by all the five sides of the pentagon. A pentagon is a five-sided polygon and a two-dimensional geometrical figure. Its name is derived from the Greek words “Penta” which means “five” and “gon” which means “angles”.

### What is the Area of Pentagon?

The area of a pentagon is the region that is covered within the sides of the pentagon. It is expressed in square units like m^{2}, cm^{2}, in^{2}, or ft^{2}, etc. A pentagon has 5 sides and the sum of its interior angles is equal to 540°, while the sum of its exterior angles equals 360°. The area of a pentagon can be calculated by various methods depending on the dimensions that are known. The different methods are given below:

**Area of Pentagon using only the side length:**In order to find the area of pentagon when only the side length is given, we use the formula:

**Area of Pentagon using the apothem:**The formula that is used to calculate the area of pentagon using the apothem is, Area = 5/2 × s × a; where ‘a’ is the apothem length, and ‘s’ is the side of the pentagon.**Area of Pentagon using the area of isosceles triangles:**In this method, the pentagon can be divided into 5 equal isosceles triangles, the area of all the 5 isosceles triangles can be calculated and added to find the area of the pentagon.

## Area of Pentagon Formula

The formula that is used to find the area of a pentagon using the apothem and side length is, Area = 5/2 × s × a. Here, “s” is the side length and “a” is the apothem of the pentagon. Observe the following pentagon to see the apothem ‘a’ and the side length ‘s’.

The area of a regular pentagon can also be calculated if only the side length “s” is known. The formula used to find the area of a regular pentagon,

where “s” is the length of one side of the regular pentagon.

## Area Formula for a Pentagon

The **Area of a Pentagon Formula** is,

A = (5 ⁄ 2) × s × a |

Where,

- “s” is the side of the Pentagon
- “a” is the apothem length

## How to Find the Area of a Regular Pentagon

A pentagon is a polygon with five straight sides. Almost all problems you’ll find in math class will cover regular pentagons, with five equal sides. There are two common ways to find the area, depending on how much information you have.

**Finding the Area from the Side Length and Apothem**

**Start with the side length and apothem.** This method works for regular pentagons, with five equal sides. Besides the side length, you’ll need the “apothem” of the pentagon. The apothem is the line from the center of the pentagon to a side, intersecting the side at a 90º right angle.

- Don’t confuse the apothem with the radius, which touches a corner (vertex) instead of a midpoint. If you only know the side length and radius, skip down to the next method instead.
- We’ll use an example pentagon with side length
**3**units and apothem**2**units.

**Divide the pentagon into five triangles.** Draw five lines from the center of the pentagon, leading to each vertex (corner). You now have five triangles.

**Calculate the area of a triangle.** Each triangle has a **base** equal to the side of the pentagon. It also has a **height** equal to the pentagon’s apothem. (Remember, the height of a triangle runs from a vertex to the opposite side, at a right angle.) To find the area of any triangle, just calculate ½ x base x height.

- In our example, area of triangle = ½ x 3 x 2 =
**3**square units.

**Multiply by five to find the total area.** We’ve divided the pentagon into five equal triangles. To find the total area, just multiply the area of one triangle by five.

- In our example, A(total pentagon) = 5 x A(triangle) = 5 x 3 =
**15**square units.

**Finding the Area from the Side Length**

**Start with just the side length.** This method only works for regular pentagons, which have five sides of equal length.

- In this example, we’ll use a pentagon with side length
**7**units.

**Divide the pentagon into five triangles.** Draw a line from the center of the pentagon to any vertex. Repeat this for every vertex. You now have five triangles, each the same size.

**Divide a triangle in half.** Draw a line from the center of the pentagon to the base of one triangle. This line should hit the base at a 90º right angle, dividing the triangle into two equal, smaller triangles.

**Label one of the smaller triangles.** We can already label one sides and one angle of the smaller triangle:

- The
**base**of the triangle is ½ the side of the pentagon. In our example, this is ½ x 7 = 3.5 units. - The
**angle**at the pentagon’s center is always 36º. (Starting with a full 360º center, you could divide it into 10 of these smaller triangles. 360 ÷ 10 = 36, so the angle at one triangle is 36º.)

**Calculate the height of the triangle.** The **height** of this triangle is the side at right angles to the pentagon’s edge, leading to the center. We can use beginning trigonometry to find the length of this side:

- In a right-angle triangle, the
**tangent**of an angle equals the length of the opposite side, divided by the length of the adjacent side. - The side opposite the 36º angle is the base of the triangle (half the pentagon’s side). The side adjacent to the 36º angle is the height of the triangle.
- tan(36º) = opposite / adjacent
- In our example, tan(36º) = 3.5 / height
- height x tan(36º) = 3.5
- height = 3.5 / tan(36º)
- height = (about)
**4.8**units.

**Find the area of the triangle.** A triangle’s area equals ½ the base x the height. (A = ½bh.) Now that you know the height, plug in these values to find the area of your small triangle.

- In our example, Area of small triangle = ½bh = ½(3.5)(4.8) = 8.4 square units.

**Multiply to find the area of the pentagon.** One of these smaller triangles covers 1/10 of the pentagon’s area. To find the total area, multiply the area of the smaller triangle by 10.

- In our example, the area of the whole pentagon = 8.4 x 10 =
**84**square units.

**Using a Formula**

**Use the perimeter and apothem.** The apothem is a line from the center of a pentagon, that hits a side at a right angle. If you are given its length, you can use this easy formula

- Area of a regular pentagon =
*pa*/2, where*p*= the perimeter and*a*= the apothem. - If you don’t know the perimeter, calculate it from the side length: p = 5s, where s is the side length.

**Use the side length.** If you only know the side length, use the following formula:

- Area of a regular pentagon = (5
*s*^{2}) / (4tan(36º)), where*s*= side length. - tan(36º) = √(5-2√5).
^{ }So if your calculator doesn’t have a “tan” function, use the formula Area = (5*s*^{2}) / (4√(5-2√5)).

**Choose a formula that uses radius only.** You can even find the area if you only know the radius. Use this formula:

- Area of a regular pentagon = (5/2)
*r*^{2}sin(72º), where*r*is the radius.

### Types of Pentagon

Any polygon has four different types: concave polygon, convex polygon, regular polygon, and irregular polygon. Similarly, the pentagon has four types. They are,

- Concave Pentagon
- Convex Pentagon
- Regular Pentagon
- Irregular Pentagon

### Concave Pentagon

A pentagon in which at least one angle is more than 180^{∘} is called a concave pentagon. In other words, if the vertices point inwards or pointing inside a pentagon, it is known as a concave pentagon.

Here, in the pentagon ABCDE, we can see that the interior angle ∠ABC is more than 180^{∘}. Hence, this is a concave pentagon.

### Convex Pentagon

A pentagon in which each angle is less than 180^{∘}. is called a convex pentagon. In other words, if the vertices point outwards in a pentagon or pointing outside is known as a convex pentagon.

### Regular Pentagon

A polygon is said to be regular if all the sides and angles of it are equal. A pentagon with all sides equal and all the angles equal is called a regular pentagon.

The pentagon ABCDE is a regular pentagon if AB=BC=CD=DE=EA and ∠A=∠B=∠C=∠D=∠E.

### Irregular Pentagon

Pentagons that aren’t regular are called irregular pentagons. In other words, a pentagon in which the sides and angles have different measures is known as an irregular pentagon.

### Interior and Exterior Angle of a Regular Pentagon

We know that, for a regular polygon of nn sides, we have

Sum of exterior angles equal to 360^{∘}.

### Perimeter of a Pentagon

The computation of the length of the boundary of any closed figure is known as its perimeter. The perimeter of a regular or irregular pentagon will be the sum of the lengths of its sides.

If a,b,c,d,e are respectively the lengths of sides of an irregular pentagon, then the perimeter of an irregular pentagon is a+b+c+d+e.

That is, perimeter p=a+b+c+d+e

If a represents the side of a regular pentagon, then the perimeter of a regular pentagon =5a.

That is, perimeter P=5a

### Apothem

A line drawn from the centre of any polygon to the mid-point of one of the sides is known as apothem.

In the above hexagon, a is the apothem.

The formula to find the apothem is based on the side length and the radius.

When the side length is given, then the apothem formula is given by

Where a= apothem length

s= length of the side of the pentagon

n= number of sides in the given polygon

When the radius is given, the apothem formula is given by

Where r= radius

n= number of sides in the given polygon

### Area of Regular Pentagon with Apothem

Assume a pentagon is having side ss and length of apothem a. The area of the pentagon is equal to ^{5}/_{2} times the length of the apothem and the side of the pentagon.

The formula to calculate the area of the pentagon is ^{5}/_{2}×s×asq.units

### Area of a Regular Pentagon without Apothem

The area of a regular pentagon can similarly be expressed in terms of length of side a.a.

The area of a regular pentagon with side length a is given by

Also, we can use another formula to find the area of the pentagon when only the side length is given. It is given by

### Area of Regular Pentagon with Radius

Let r be the length of the radius of a regular pentagon. Then the area of a regular pentagon is given by

## Calculation of Area of Pentagon

Let us calculate the area of a pentagon using the basic formula, Area = 5/2 × s × a, where ‘a’ is the apothem and ‘s’ is the side length.

**Example:** Find the area of the pentagon whose side length is 18 units and the length of apothem is 5 units.

**Solution: **Given that s = 18 units and a = 5 units

Thus, the area of the pentagon, A = 5/2 × s × a

⇒ A = 5/2 × 18 × 5

⇒ A = 225 square units

The area of the pentagon is 225 square units.

### Solved Examples on Area of Pentagon

**Example 1:** Find the area of a pentagon if the length of the side of the pentagon is 10 in, and the length of the apothem is 6.88 in.

**Solution: **Given s = 10 in and a = 6.88 in.

Using the formula for the area of pentagon,

A = 5/2 × s × a

⇒ A = 5/2 × 10 × 6.88

⇒ A = 172 in^{2}

**Answer:** The area of the pentagon is 172 square inches.

**Example 2:** Jack was given the area of a pentagon as 500 units square with a side length of 17 units. Can you help him find the length of the apothem of the pentagon?

**Solution:** Given A = 500 units square and s = 17 units

Using the formula for the area of Pentagon,

A = 5/2 × s × a

⇒ 500 = 5/2 × 17 × a

⇒ a = 11.76 units

**Answer: **The length of the apothem of the pentagon is 11.76 units.

**Question 1: **Find the area of a pentagon of side 10 cm and apothem length 5 cm.

**Solution: **Given,

s = 10 cm

a = 5 cm

Area of a pentagon = A = (5 ⁄ 2) × s × a

= (5 ⁄ 2) × 10 × 5 cm^{2}

= 125 cm^{2}.

**Question 2:** Find the area of a regular pentagon, if the side length is 3cm.

Solution: Given, side length, s = 3 cm

Area of a regular pentagon=

By putting the value of s = 3 cm, we get;

Area = 15.48 cm^{2}

** Q.1. Find the area of the given regular pentagon whose side measure is 4cm.Ans:** We know that the area of a regular pentagon with side measure a units is given by

** Q.2. Find the area of the given regular pentagon whose side measure is 3cm.Ans: **We know that the area of a regular pentagon with side measure aa units is given by

Therefore, the area of a regular pentagon with a side of 3cm is 15.484cm2.

*Q.3. Find the area and perimeter of a regular pentagon whose side is *6cm, *and apothem length is *5cm.* Ans: *Given: side measure of regular pentagon s=6cm

The measure of apothem a=5cm

⇒A=75 cm2

Now, we know that the perimeter of a regular pentagon with side length aa units is 5a.

Therefore, the perimeter of the given pentagon =5×6cm

=30cm

Therefore, the area of the given pentagon is 75cm2 and the perimeter is 30cm.

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

Find the area of the pentagon shown above.

**Possible Answers:**

340 square units

390 square units

380 square units

320 square units

300 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×length

A=20×15=300

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:

A=300+40+40=380

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

Find the area of the pentagon shown above.

**Possible Answers:**

672 square units

502 square units

416 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×length

A=26×16=416

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:

A=416+78+78=572

### Example Question #3 : How To Find The Area Of A Pentagon

A regular pentagon has a side length of 9 inches and an apothem length of 6.2 inches. Find the area of the pentagon.

**Possible Answers:**

36 inches2

139.5 inches2

118 inches2

99 inches2

**Correct answer:**

139.5 inches2

By definition a regular pentagon must have 5 equal sides and 5 equivalent interior angles. Since we are told that this pentagon has a side length of 9 inches, all of the sides must have a length of 9 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 5 equivalent interior triangles. Each triangle will have a base of 9 and a height of 6.2.

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:

27.9×5=139.5

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

A regular pentagon has a side length of 14 and an apothem length of 9.6. Find the area of the pentagon.

**Possible Answers:**

67.2 square units

84 square units

82 square units

336 square units

200 square units

**Correct answer:**

336 square units

By definition a regular pentagon must have 5 equal sides and 5 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 5 equivalent interior triangles. Each triangle will have a base of 14 and a height of 9.6.

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

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

67.2×5=336

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

A regular pentagon has a perimeter of 60 yards and an apothem length of 8.3 yards. Find the area of the pentagon.

**Possible Answers:**

48 yards2

249 yards2

96 yards2

200 yards2

248 yards2

**Correct answer:**

249 yards2

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

p=5(s)

60=5(s)

s=60/5=12

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

Note: a regular pentagon must have 5 equal sides and 5 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 5 equivalent interior triangles. Each triangle will have a base of 12 and a height of 8.3.

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

Thus, the area of the entire pentagon is:

A=49.8×5=249

### Example Question #6 : How To Find The Area Of A Pentagon

A regular pentagon has a side length of 15 and an apothem length of 10.3. Find the area of the pentagon.

**Possible Answers:**

383.5 square units

306.25 square units

400.5 square units

270 square units

386.25 square units

**Correct answer:**

386.25 square units

By definition a regular pentagon must have 5 equal sides and 5 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 5 equivalent interior triangles. Each triangle will have a base of 15 and a height of 10.3.

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:

A=77.25×5=386.25

### Example Question #7 : How To Find The Area Of A Pentagon

A regular pentagon has a perimeter of 50 and an apothem length of 7. Find the area of the pentagon.

**Possible Answers:**

70 square units

135 square units

170 square units

175 square units

**Correct answer:**

175 square units

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

p=5(s)

50=5(s)

s=50/5=10

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

Note: a regular pentagon must have 5 equal sides and 5 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 5 equivalent interior triangles. Each triangle will have a base of 10 and a height of 7.

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:

35×5=175

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

A regular pentagon has a side length of 25 and an apothem length of 17.2. Find the area of the pentagon.

**Possible Answers:**

900 square units

215 square units

1,350 square units

725 square units

1,075 square units

**Correct answer:**

1,075 square units

By definition a regular pentagon must have 5 equal sides and 5 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 5 equivalent interior triangles. Each triangle will have a base of 25 and a height of 17.2.

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:

215×5=1,075

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

A regular pentagon has a side length of 7 and an apothem length of 4.8. Find the area of the pentagon.

**Possible Answers:**

68 sq. units

133.6 sq. units

84 sq. units

16.8 sq. units

33.6 sq. units

**Correct answer:**

84 sq. units

By definition a regular pentagon must have 5 equal sides and 5 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 5 equivalent interior triangles. Each triangle will have a base of 7 and a height of 4.8.

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

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

16.8×5=84

### Example Question #10 : How To Find The Area Of A Pentagon

A regular pentagon has a perimeter of 65 and an apothem length of 8.9. Find the area of the pentagon.

**Possible Answers:**

289.25

275.85

280.50

57.85

**Correct answer:**

289.25

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

p=5(s)

65=5(s)

s=655=13

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

Note: a regular pentagon must have 5 equal sides and 5 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 5 equivalent interior triangles. Each triangle will have a base of 13 and a height of 8.9.

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

Thus, the area of the entire pentagon is:

A=57.85×5=289.25

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

Find the area of the pentagon shown above.

**Possible Answers:**

189 sq. units

198 sq. units

180 sq. units

54 sq. units

184 sq. units

**Correct answer:**

198 sq. 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:

A=width×length

A=18×8=144

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:

A=144+27+27=198

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

A regular pentagon has a perimeter of 145 and an apothem length of 20. Find the area of the pentagon.

**Possible Answers:**

1,580 sq. units

1,450 sq. units

1,500 sq. units

580 sq. units

450 sq. units

**Correct answer:**

1,450 sq. units

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

p=5(s)

145=5(s)

s=145/5=29

Note: a regular pentagon must have 5 equal sides and 5 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 5 equivalent interior triangles. Each triangle will have a base of 29 and a height of 20.

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

Thus, the area of the entire pentagon is:

A=290×5=1,450

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

Find the area of the regular pentagon.

**Possible Answers:**

15

None of these

12

18

**Correct answer:**

15

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

Area of Regular Polygon=1/2(apothem)(perimeter)

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

Perimeter=5(side length)

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

Perimeter=5(3)

Perimeter=15

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

Area of Pentagon=^{1}/_{2}(2)(15)

Area of Pentagon=15

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

Find the area of the regular pentagon.

**Possible Answers:**

400

380

390

420

**Correct answer:**

420

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

Area of Regular Polygon=1/2(apothem)(perimeter)

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

Perimeter=5(side length)

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

Perimeter=5(14)

Perimeter=70

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

Area of Pentagon=^{1}/_{2}(12)(70)

Area of Pentagon=420

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

Find the area of the regular pentagon.

**Possible Answers:**

65

55

50

60

**Correct answer:**

60

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

Area of Regular Polygon=1/2(apothem)(perimeter)

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

Perimeter=5(side length)

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

Perimeter=5(8)

Perimeter=40

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

Area of Pentagon=1/2(5)(40)

Area of Pentagon=60

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

Find the area of the regular pentagon.

**Possible Answers:**

88

62

78

70

**Correct answer:**

70

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

Area of Regular Polygon=1/2(apothem)(perimeter)

Perimeter=5(side length)

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

Perimeter=5(7)

Perimeter=35

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

Area of Pentagon=1/2(4)(35)

Area of Pentagon=70

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

Find the area of the regular pentagon.

**Possible Answers:**

420

630

700

560

**Correct answer:**

560

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

Area of Regular Polygon=1/2(apothem)(perimeter)

Perimeter=5(side length)

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

Perimeter=5(16)

Perimeter=80

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

Area of Pentagon=1/2(14)(80)

Area of Pentagon=560

### Example Question #18 : How To Find The Area Of A Pentagon

Find the area of the regular pentagon.

**Possible Answers:**

325

355

340

370

**Correct answer:**

325

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

Area of Regular Polygon=1/2(apothem)(perimeter)

Perimeter=5(side length)

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

Perimeter=5(13)

Perimeter=65

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

Area of Pentagon=1/2(10)(65)

Area of Pentagon=325

### Example Question #19 : How To Find The Area Of A Pentagon

Find the area of the regular pentagon.

**Possible Answers:**

1220

1260

1180

1300

**Correct answer:**

1260

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

Area of Regular Polygon=1/2(apothem)(perimeter)

Perimeter=5(side length)

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

Perimeter=5(24)

Perimeter=120

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

Area of Pentagon=^{1}/_{2}(21)(120)

Area of Pentagon=1260

### Example Question #20 : How To Find The Area Of A Pentagon

Find the area of the regular pentagon.

**Possible Answers:**

1300

1600

1900

1000

**Correct answer:**

1600

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

Area of Regular Polygon=12(apothem)(perimeter)

Perimeter=5(side length)

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

Perimeter=5(20)

Perimeter=100

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

Area of Pentagon=1/2(16)(100)

Area of Pentagon=1600

### Example Question #25 : How To Find The Area Of A Pentagon

Each side of this pentagon has a length of 5cm.

Solve for the area of the pentagon.

**Possible Answers:**

43cm2

25cm2

32.7cm2

21.5cm2

35.5cm2

**Correct answer:**

43cm2

The formula for area of a pentagon is A=5⋅(^{1}/_{2}⋅s⋅h), with s representing the length of one side and h representing the apothem.

To find the apothem, we can convert our one pentagon into five triangles and solve for the height of the triangle:

Each of these triangles have angle measures of 72$∘$54$∘$54$∘$, with 72$∘$being the angle oriented around the vertex. This is because the polygon has been divided into five triangles and 360/5=72.

### Example Question #26 : How To Find The Area Of A Pentagon

Each side of this regular pentagon has a length of 7.5cm. Solve for the area of the pentagon. Round to the nearest tenth.

**Possible Answers:**

140.6cm2

35.5cm2

48.4cm2

96.8cm2

37.5cm2

**Correct answer:**

96.8cm2

When given the value of one side of a regular pentagon, we can assume all sides to be of equal length and we can use this formula to calculate area:

For thi formula, s represents the length of one side while N represents the number of sides. Therefore, we would plug in the values as such:

### Example Question #27 : How To Find The Area Of A Pentagon

A regular pentagon has a side length of 17cm. Find the area rounded to the nearest tenth.

**Possible Answers:**

497.2cm2

99.4cm2

29.2cm2

225.7cm2

383.6cm2

**Correct answer:**

497.2cm2

We can use the following equation to solve for the area of a regular polygon with s representing side length and N representing number of sides:

### Example Question #28 : How To Find The Area Of A Pentagon

In the figure below, find the area of the pentagon if the length of AF is one-fourth the length of FD.

**Possible Answers:**

230

130

190

150

**Correct answer:**

230

Start by figuring out the lengths of AF and FD.

Let x be the length of AF. From the question, we then write the length of FD as 4x.

From the figure, you should see that AF+FD=AD=BC.

Plug in the variables and solve for x.

x+4x=20

5x=20

x=4

So then the length of AF must be 4, and the length of FD must be 16.

Now, to find the area of the pentagon, notice that we can break the shape down to one rectangle and two right triangles.

Next, find the area of rectangle ABCD.

Area of rectangle=length×width

Area of ABCD=9×20=180

Next, find the area of the right triangles.

Area of Triangle=1/2(base)(height)

For triangle AFE,

Area=1/2(4)(5)=10

For triangle EFD,

Area=1/2(16)(5)=40

Add up the component areas to find the area of the pentagon.

Area of Pentagon=180+10+40=230

## FAQs on Area of Pentagon

### What is the Area of a Pentagon?

The area of a pentagon is the region that is covered by all the sides of the pentagon. It can be calculated by various methods according to the dimensions given. The basic formula that is used to find the area of a pentagon is, Area = 5/2 × s × a; where ‘s’ is the side length and ‘a’ is the apothem.

### What is the Unit of Area of Pentagon?

The unit of area of the pentagon is expressed in square units, for example, m^{2}, cm^{2}, in^{2}, or ft^{2}, etc.

### How to find the Area of Pentagon using Apothem?

If the apothem of the pentagon is given, its area can be calculated using the formula: Area = 5/2 × s × a; where ‘s’ is the side length and ‘a’ is the apothem.

### How to find the Area of Pentagon using Side Length?

In case of a regular pentagon, if the side length is known then the area of the pentagon can be calculated by the formula:

where “s” is the length of one side of the regular pentagon. However, if it is not a regular pentagon, then the area can be calculated using the formula, Area = 5/2 × s × a; where ‘s’ is the side length and ‘a’ is the apothem.

### What is the Area of Pentagon Formula?

The basic formula that is used to find the area of a pentagon is, Area = 5/2 × s × a; where ‘s’ is the length of the side of the pentagon and ‘a’ is the apothem of a pentagon.

### How to Find the Area of Pentagon?

We can find the area of the pentagon using the following steps:

- Step 1: Find the length of the side of the pentagon.
- Step 2: Identify the apothem length of the pentagon.
- Step 3: Use the formula, Area = 5/2 × s × a; where ‘s’ is the side length and ‘a’ is the apothem.
- Step 4: This will give the area of the pentagon and we represent the answer in square units.

### How To Find the Area of Regular Pentagon?

We can find the area of the regular pentagon using the following steps:

- Step 1: Find the length of the side of the pentagon.
- Step 2: Find the area of regular pentagon using the formula

- where “s” is the length of one side of a regular pentagon.
- Step 4: This will give the area of the regular pentagon and represent the answer in square units.

### What Happens to the Area of Pentagon if the Length of the Side is Tripled?

The area of the pentagon is tripled if the length of the side is tripled as the “s” is substituted to “3s” in the formula A = 5/2 × s × a = 5/2 × (3s) × a = 3 × (5/2 × s × a) which gives three times the original area of the pentagon.

### What Happens to the Area of Pentagon If the Length of Apothem is Halved?

The area of the pentagon is halved if the length of the side is tripled as the “a” is substituted to “a/2” in the formula A = 5/2 × s × a = 5/2 × s × (a/2) = (1/2) × (5/2 × s × a) which gives half the original area of the pentagon.

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