Surface Area of Pentagonal Prism
The surface area of a prism is the amount of space occupied by the surface of a prism. A pentagonal prism is a type of heptahedron having 7 faces, 15 edges, and 10 vertices. It is a 3D shape that has a pentagon base and a pentagon top. Like all threedimensional shapes, you will learn how to calculate the surface area of a pentagonal prism. Let's learn more about its formula using solved examples.
1.  What is the Surface Area of Pentagonal Prism? 
2.  Formula of Surface Area of Pentagonal Prism 
3.  Derivation of Surface Area of Pentagonal Prism 
4.  FAQs on Surface Area of Pentagonal Prism 
What is the Surface Area of Pentagonal Prism?
The space occupied by the surface of a pentagonal prism is known as the surface area of a pentagonal prism. Since it has a flat base, thus it has a total surface area as well as a curved/lateral surface area. A pentagonal prism has 15 edges, 10 vertices, and 7 faces (2 pentagonal and 5 rectangular). We can classify pentagonal prisms on the basis of their rectangular and pentagonal faces.
 Regular pentagonal prisms: A regular pentagonal prism is a pentagonal prism whose bases have all sides equal, the side faces are rectangles, and the angles between the sides of the base are 108 degrees. In other words, a regular pentagonal prism has congruent rectangular faces.
 Right pentagonal prisms: A right pentagonal prism has congruent and parallel pentagonal faces and rectangular faces are perpendicular to the pentagon faces.
 Oblique pentagonal prisms: The pentagon faces of a pentagonal prism are not exactly aligned on each other and the rectangular faces are not perpendicular to the pentagon faces.
Formula of Surface Area of Pentagonal Prism
As a pentagonal prism has a lateral surface, thus we can express its lateral surface area as well as total surface area. A pentagonal prism has two kinds of surface area:
 Total Surface Area
 Curved/Lateral Surface Area
The surface area of a pentagonal prism gives the area of each face of the prism. If the apothem length of the prism is "a", the base length of the prism is "b" and the height of the prism is "h", the surface area of a pentagonal prism is given as:
 Total Surface Area, T = 5ab + 5bh sq. units.
 Lateral Surface Area, L = Ph = 5bh sq. units
The pentagonal prism formula for calculating surface area remains the same for all kinds of pentagonal prisms.
 In the case of a regular pentagonal prism, both the pentagonal sides are of equal length, and the five rectangular sides with bases being perpendicular to each other.
 In the case of a right pentagonal prism, the bases are perpendicular to each other.
 In the case of an oblique pentagonal prism, the bases are not perpendicular to each other. Thus, the perpendicular drawn from one vertex of one base to the other base of the prism will be taken as its height.
Derivation of Surface Area of Pentagonal Prism
Let us take a pentagonal prism, apothem length of the prism is "a", the base length of the prism is "b", and height of the prism is "h. In order to determine we cut the pentagonal prism into its net and open it. In general, the lateral surface area of a prism is equal to five times the product of its base and height. Thus, the lateral surface area of a pentagonal prism can be given by finding the perimeter of the base of the prism multiplied by its height. Therefore, the lateral surface area of the pentagonal prism, LSA = 5b × h = 5bh
The total surface area of pentagonal prism = 5 × area of the base of pentagonal prism + lateral surface area of the pentagonal prism
⇒ Total surface area of pentagonal prism = 5ab + 5bh
Therefore, the total surface area of the pentagonal prism, TSA = 5ab + 5bh
Example: Find the total surface area of the pentagonal prism and lateral surface area of the pentagonal whose base length is given as 20 units, apothem length is 15 units and height is 15 units.
Solution: As we know, the total surface area of the pentagonal prism is 5ab + 5bh, and the lateral surface area of a pentagonal prism is 5bh.
Given that: base length = 20 units, apothem length = 15 units, and height =15 units.
Thus, the lateral surface area of pentagonal prism = 5bh = 5 × 20 × 15 = 1500 sq units
The total surface area of pentagonal prism = 5ab + 5bh = 5ab + LSA = (5 × 15 × 20) + 1500 = 1500 + 1500 = 3000 sq units.
Therefore, the lateral surface area of the pentagonal prism is 1500 sq units and the total surface area of the pentagonal prism is 3000 sq units.
Solved Examples on Surface Area of Pentagonal Prism

Example 1: Find the lateral surface area of a regular pentagonal prism if the perimeter of the base = 270 inches and height = 75 inches.
Solution: The lateral surface area of pentagonal prism is, L = Ph, where P is the perimeter of the base and h is the height of the prism
Here, P = 270 in, h = 75 in
⇒ LSA = Ph = (270)×(75) = 20,250 sq. inchesAnswer: Therefore, the lateral surface area of the pentagonal prism is 20,250 square inches.

Example 2: Find the height of the pentagonal prism if the total surface area of the pentagonal prism is 90 sq. yd., apothem length and base length are 4 yds. and 2 yds. respectively.
Solution: Given: Total surface of pentagonal prism, TSA = 90 sq yd, a = 4 yd and b = 2 yd
We know that the total surface area of a pentagonal prism = 5ab + 5bh = 5b(a+h) = 90
⇒ 10(4+h) = 90 ⇒ 4 + h = 90/10Thus, h = 9  4
⇒ h = 5 ydsAnswer: Therefore, the height of the pentagonal prism is 5 yds.
FAQs on Surface Area of Pentagonal Prism
What is the Total Surface Area of a Pentagonal Prism in Geometry?
The total surface area of the pentagonal prism is the sum of the area of the bases of the pentagonal prism and the lateral surface area of the pentagonal prism. Therefore, the total surface area of the pentagonal prism is 5ab + 5bh.
How Do You Find the Surface Area of a Pentagonal Prism?
To find the surface area of a pentagonal prism, follow the steps given below:
 Step 1: Identify the apothem length a and the base length b of the given pentagonal prism.
 Step 2: Find the height h of the pentagonal prism, which is the total height of the prism.
 Step 3: Put the values of a, b, and h in the surface area of the pentagonal prism formula, TSA = 5ab + 5bh
 Step 4: Write the value so obtained in square units.
What is the Formula for the Surface Area of a Pentagonal Prism?
In general, the lateral surface area of the pentagonal prism is equal to five times the product of its base and height. Thus, the lateral surface area of a pentagonal prism can be given by finding the perimeter of the base of the prism multiplied by its height. Therefore, the lateral surface area of the pentagonal prism, LSA = 5b × h = 5bh. The total surface area of pentagonal prism = 2 × area of the base of pentagonal prism + lateral surface area of the pentagonal prism. Therefore, the total surface area of pentagonal prism = 5ab + 5bh.
How Do You Find the Lateral Surface Area of a Pentagonal Prism?
The lateral surface area of the regular pentagonal prism includes the area with 5 rectangular faces, leaving two pentagonal faces. Follow the steps given below to find the lateral surface area of the pentagonal prism:
 Step 1: Find the perimeter of the base of the pentagonal prism, that is 5a (a=apothem length)
 Step 2: Find the height h of the pentagonal prism.
 Step 3: Multiply the answer in step 1 with the answer in step 2.
 Step 4: Write the value with the appropriate unit. (in square units)
Thus, the lateral surface area of the pentagonal prism equals the product of the perimeter of one base and height, that is Ph or 5bh.
What Happens to the Lateral Surface Area of Pentagonal Prism If Its Height is Doubled?
We know that the lateral surface area of a pentagonal prism, LSA is 5bh. Thus, if its height will be doubled, then the lateral surface area will be 5b(2h) = 2(5bh) = 2LSA. Therefore, we can conclude that the lateral surface will also get doubled.
What Happens to the Total Surface Area of Pentagonal Prism If Length of Apothem is Doubled?
We know that the total surface area of a pentagonal prism, TSA is 5ab + 5bh. Thus, if its apothem length will be doubled, then the lateral surface area will increase but it will certainly not get doubled. Therefore, we can conclude that the total surface will increase if the length of the apothem is doubled.
What Happens to the Lateral Surface Area of Pentagonal Prism If Its Base Length is Doubled?
The formula for calculating the lateral surface area of a pentagonal prism, LSA is 5bh. Thus, if its base length will be doubled, then the lateral surface area will be 5h(2b) = 2(5bh) = 2LSA. Therefore, we can conclude that the lateral surface will also get doubled.