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 fivesided polygon and a twodimensional geometrical figure. Its name is derived from the Greek words 'Penta' which means 'five' and 'gon' which means 'angles'. In this lesson, we will learn how to find the area of pentagon with solved examples and practice questions.
1.  What is the Area of Pentagon? 
2.  Area of Pentagon Formula 
3.  How to Find the Area of Pentagon? 
4.  Area of Regular Pentagon 
5.  Area of Irregular Pentagon 
6.  FAQs on Area of Pentagon 
What is the Area of Pentagon?
The area of a pentagon is the space that is covered within the sides of the pentagon. It can be calculated by various methods depending on the dimensions that are known. It also depends on the type of pentagon. For example, if it is a regular pentagon, then the area can be calculated with the help of one single formula, but if it is an irregular pentagon, then we need to split it into different polygons and add their areas to get the area of the pentagon. The area of a pentagon is expressed in square units like m^{2}, cm^{2}, in^{2}, ft^{2}, and so on.
Area of Pentagon Formula
The formula that is used to find the area of a pentagon varies according to the type of pentagon. The area of a pentagon formula that is commonly used to find the area of a regular pentagon is,
Area of pentagon = 1/2 × p × a
Here, 'p' is the perimeter and 'a' is the apothem of the pentagon. Observe the following pentagon to see the apothem 'a' and the side length 's'.
How to Find the Area of Pentagon?
The area of a pentagon can be calculated using different methods and formulas depending on the values that are given and also on the kind of pentagon.
Area of Regular Pentagon
The area of a regular pentagon can be calculated if only the side length 's' is known. The formula used to find the area of a regular pentagon, \(A = \frac{1}{4}\sqrt{5(5+2\sqrt{5})}s^{2}\) where 's' is the length of one side of the regular pentagon.
Example: Find the area of a pentagon whose side is 7 units.
Solution: Let us use the formula for the area of a regular pentagon = \(A = \frac{1}{4}\sqrt{5(5+2\sqrt{5})}s^{2}\); where s = 7. After substituting the value of s = 7, we get, \(A = \frac{1}{4}\sqrt{5(5+2\sqrt{5})}7^{2}\) = 84.3 square units.
Area of Pentagon with Apothem
The area of a pentagon can be calculated if the side and apothem is given. The formula that is commonly used to find the area of any regular polygon using the apothem and side is, Area of regular polygon = 1/2 × perimeter of polygon × apothem.
So, Area of regular pentagon = 1/2 × p × a; where 'p' is the perimeter of the pentagon and 'a' is the apothem of the pentagon. Let us understand this with an example.
Example: Find the area of a regular 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, let us first find the perimeter of the pentagon. Perimeter of pentagon = 5 × side length = 5 × 18 = 90 units. Now, let us substitute these values in the formula
Area of the pentagon, A = 1/2 × p × a; where p = 90, a = 5
⇒ A = 1/2 × 90 × 5
⇒ A = 225 square units
Therefore, the area of the pentagon is 225 square units.
Area of Irregular Pentagon
The area of an irregular pentagon can be calculated by dividing the pentagon into other smaller polygons. Then, the area of these polygons is calculated and added together to get the area of the pentagon. Let us understand this with an example.
Example: Find the area of a pentagon ABCDE whose sides are given as AB = 5 cm, BC = 4 cm, CD = 8 cm, DE = 4 cm, EA = 5 cm.
Solution: We can find the area of the pentagon using the following steps:
 Step 1: First, we will split the pentagon into a triangle ABE and a rectangle BCDE.
 Step 2: Then, we will find the area of triangle ABE and the area of rectangle BCDE. The area of triangle ABE can be calculated using Heron's formula since we know the 3 sides of the triangle. AB = 5 cm, BE = 8 cm, AE = 5 cm. So, area of triangle ABE = √[s(sa)(sb)(sc)], where s = Perimeter/2 = (a + b + c)/2 . Now, s = (5 + 8 + 5)/2 = 18/2 = 9. After substituting the values in the formula, we get, Area of Δ ABE = √[9(95)(98)(95)] = √(9 × 4 × 1 × 4) = √144 = 12 cm^{2}.
 Step 3: The area of a rectangle = Length × Width. Here, length (CD) = 8 cm, Width (BC) = 4 cm. So, area of rectangle BCDE = 8 × 4 = 32 cm^{2}.
 Step 4: Add the areas of the triangle and the rectangle. Therefore, the Area of pentagon ABCDE = Area of triangle ABE + Area of rectangle BCDE = 12 + 32 = 44 cm^{2}.
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Examples on Area of Pentagon

Example 1: Find the area of a regular pentagon if the length of the side of the pentagon is 10 inches, and the length of the apothem is 6.88 inches.
Solution: Given, s = 10; and a = 6.88, The formula to find the area of a pentagon with apothem is, Area of pentagon = 1/2 × perimeter × apothem. So, let us find the perimeter of the pentagon. Perimeter of the pentagon = 5 × side length = 5 × 10 = 50 inches. Now, let us substitute these values in the formula.
Area of pentagon = 1/2 × p × a
⇒ A = 1/2 × 50 × 6.88
⇒ A = 172 in^{2}Answer: The area of the pentagon is 172 square inches.

Example 2: If the area of a pentagon is 500 units square with a side length of 17 units, find the length of the apothem of the pentagon.
Solution: Given A = 500 units square and s = 17 units. First, let us find the perimeter of the pentagon using the formula, Perimeter of pentagon = 5 × side length = 5 × 17 = 85 units. Now, let us substitute these values in the following formula to find the length of the apothem.
Using the formula for the area of Pentagon,
Area of pentagon = 1/2 × p × a
⇒ 500 = 1/2 × 85 × a
⇒ apothem = 11.76 unitsAnswer: The length of the apothem of the pentagon is 11.76 units.
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Example 3: Find the area of the pentagon whose side length is 5 units and the length of apothem is 6 units.
Solution:
Given, s = 5; and a = 6, The formula to find the area of a pentagon with apothem is, Area of pentagon = 1/2 × perimeter × apothem. So, let us find the perimeter of the pentagon. Perimeter of the pentagon = 5 × side length = 5 × 25 = 125 units. Now, let us substitute these values in the formula.
Area of pentagon = 1/2 × p × a
⇒ A = 1/2 × 125 × 6
⇒ A = 375 unit^{2}Answer: The area of the pentagon is 375 square units.
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 regular pentagon is, Area = 1/2 × perimeter of the pentagon × apothem.
What is the Unit of Area of Pentagon?
The area of the pentagon is expressed in square units, for example, m^{2}, cm^{2}, in^{2}, or ft^{2}, and so on.
How to find the Area of Pentagon with Apothem?
If the apothem of the regular pentagon is given, then its area can be calculated using the formula: Area of pentagon = 1/2 × p × a; where 'p' is the perimeter of the pentagon and 'a' is the apothem. Since it is a regular pentagon, the perimeter can be calculated with the formula, Perimeter = 5 × side, and then its value can be used in the formula.
How to find the Area of Pentagon with Side Length?
In case of a regular pentagon, if only the side length is known then the area of the pentagon can be calculated by the formula: Area = \(\frac{1}{4}\sqrt{5(5+2\sqrt{5})}s^{2}\) where 's' is the length of one side of the regular pentagon. However, if the side length and the apothem is given, then the area can be calculated using the formula, Area = 1/2 × perimeter of pentagon × apothem.
What is the Formula of Area of Pentagon?
The formula that is used to find the area of a pentagon varies according to the type of pentagon. The basic formula for the area of a regular pentagon is, Area of pentagon = 1/2 × p × a; where 'p' is the perimeter of the pentagon and 'a' is the apothem of a pentagon. However, there is no defined formula for the area of an irregular pentagon. In this case, the irregular pentagon is split into different polygons accordingly and then their areas are added to get the area of the pentagon.
How to Find the Area of Regular Pentagon?
The area of a regular pentagon can be calculated according to the given dimensions.
 When only the side length is given, then the formula that is used to find the area of a regular pentagon, is \(A = \frac{1}{4}\sqrt{5(5+2\sqrt{5})}s^{2}\) where 's' is the length of one side of the regular pentagon.
 When the side length and apothem is given, then the area can be calculated using the formula, Area = 1/2 × perimeter of pentagon × apothem.
How to Find the Area of Irregular Pentagon?
The area of an irregular pentagon can be calculated by dividing the pentagon into other smaller polygons. Then, the area of these polygons is calculated and added together to get the area of the pentagon.
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