✅ Area of a Pentagon Formula ⭐️⭐️⭐️⭐️⭐

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 m², cm², in², or ft², 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 = (5s²) / (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 = (5s²) / (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)r²sin(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,

1. Concave Pentagon
2. Convex Pentagon
3. Regular Pentagon
4. 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

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 cm²

= 125 cm².

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 cm²

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

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, m², cm², in², or ft², 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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