Surface Area of a Prism Formula

A three dimensional solid prism's surface area is determined by the form of its base. A prism's surface area is the entire area filled by the prism's faces. A prism is a flat-faced polyhedron. It lacks curves. In this article, we are going to discuss the surface area of different types of prisms and various problems based on them.

What is the Surface Area of Prism? 

The amount of total space filled by the flat faces of a prism is referred to as its surface area of a prism. Calculating the entire space occupied by all the faces of that particular type of prism, or the sum of the areas of all faces (or surfaces) on a 3D plane, is how you find the surface area of a prism.

Surface Area of a Prism Formula - Derivation & Solved Examples

A prism is a polyhedron that has two congruent parallel bases joined by parallelogram or rectangular (in the case of right prisms) faces laterally. The number of lateral surfaces is equal to the number of sides of the base polygon. For an $n$ sided base polygon, the prism will have $n$ rectangular faces. Therefore the total number of surfaces for a prism with an n sided polygonal base is $n + 2$. 

Consider such a prism whose length is $h$ and whose side length and apothem of its base are respectively $b$ and $a$. In general, the lateral surface area of a prism is equal to the product of the length of the prism and the perimeter of its base.

$\text{LSA} = {\text{perimeter}}\,{\text{of}}\,{\text{base}} \times h$

For a regular right prism, we have the following equations, 

$\text{TSA} = 2 \times \text{BA} + \text{LSA} \\ $

$\Rightarrow  \text{TSA} = 2 \times BA + n \times \text{area of one rectangular face} \\ $

$\Rightarrow  \text{TSA} = 2 \times \dfrac{n}{2} \times a \times b + n  \times h  \times b \\ $

$\therefore \text{TSA} =nb(a + h)$

Here, 

$\text{TSA}$ = Total surface area

$\text{BA}$ = Base area

$\text{LSA}$ = Lateral surface area

The following diagram shows the case of a triangular prism.

(Image will be uploaded soon)

$\text{TSA} = 2 \times BA + LSA = 2 \times \dfrac{1}{2} \times b \times h + l \times \left( {a + b + c} \right) = bh + l\left( {a + b + c} \right)$ 

In the case of a regular triangular prism, $a = b = c$. Therefore, we have the following formula for its surface area.

$\text{TSA} = bh + 3lb = b\left( {h + 3l} \right)$ 

Solved Problems

Let’s look at an example to see how to apply these formulae.

1. The two bases of a triangular prism are equilateral triangles of side length $4\,cm$. If its length is $6\,cm$, then find its total surface area.

Solution:

Given, 

No. of sides (n) = 3, 

Side length (b) = 4 cm, 

Height of the prism (h) = 6 cm, 

To Find: Total surface area of the prism, $\text{TSA}=?$

First we have to apply the base surface area formula and after that we have to apply the formula of total surface area of the prism.

The base area of the triangular prism is written as;

$\text{BA} = \dfrac{{\sqrt 3 }}{4}{b^2}\\$

$\Rightarrow \text{BA} = \dfrac{{\sqrt 3 }}{4}{4^2} \\$

$\Rightarrow \text{BA}= 4\sqrt 3 \,c{m^2} \\ $ 

$\Rightarrow \text{TSA} = 2 \times \text{BA} + n \times \text{area of one rectangular face}$

$\Rightarrow \text{TSA} = 2 \times 4 \sqrt{3} + 3 \times 4 \times 6 \\ $

$\Rightarrow \text{TSA} = 72 + 8 \sqrt{3} \\$

$\therefore \text{TSA} = 8(9 + \sqrt{3})\,cm^{2}\\$

Therefore, the total surface area of the prism will be 8(9 + \[\sqrt{3}\]) \[cm^{2}\]. 

Why don’t you try using these formulas to solve the following problem?

2. A regular hexagonal prism is $12\,cm$ long and the perimeter of its hexagonal base is $30\,cm$. Find its lateral surface area.

Options:

(a) $15\left( {12 + 15\sqrt 3 } \right)\,c{m^2}$ 

(b) $15\left( {24 + 5\sqrt 3 } \right)\,c{m^2}$

(c) $12\left( {25 + 5\sqrt 3 } \right)\,c{m^2}$

(d) $10\left( {24 + 5\sqrt 3 } \right)\,c{m^2}$

Answer: (b)

Solution:

Given, 

No. of sides (n) = 6, as it is a hexagon,

Height of the prism (h) = 12 cm, 

Perimeter of the base (nb) = 30 cm, 

To find: Total surface area of hexagonal prism, $\text{TSA}=?$

In order to solve this problem, first we have to apply the formula of perimeter and then base area formula and finally the total surface area of the hexagonal prism.

${\text{Perimeter}}\,{\text{of}}\,{\text{base}} = n \times b$

$\Rightarrow 6 \times b = 30 \\$ 

$\Rightarrow b = 5\,cm \\ $ 

Now the base area ($\text{BA}$) of the hexagonal prism is;

$\Rightarrow \text{BA} = \dfrac{{3\sqrt 3 }}{2}{b^2} \\$

$\Rightarrow \text{BA} = \dfrac{{3\sqrt 3 \times {5^2}}}{2} \\$

$\Rightarrow \text{BA} = \dfrac{{75\sqrt 3 }}{2}\,c{m^2}\\$ 

The lateral surface area ($\text{LSA}$) of the hexagonal prism is;

$\Rightarrow \text{LSA} = {\text{perimeter}}\,{\text{of}}\,{\text{base}} \times h = 30 \times 12 \\ $ 

$\Rightarrow \text{LSA} = 360\,c{m^2} \\ $ 

$\Rightarrow \text{LSA} = 2 \times \text{BA} + \text{LSA} \\ $ 

$\Rightarrow \text{LSA} = 2 \times \dfrac{{75\sqrt 3 }}{2} + 360 \\$

$\Rightarrow \text{LSA}= 360 + 75\sqrt 3 \\$

$\therefore \text{LSA} = 15\left( {24 + 5\sqrt 3 } \right)\,c{m^2}$

Therefore, the lateral surface area of the hexagon is $15\left( {24 + 5\sqrt 3 } \right)\,c{m^2}$.

Summary

In this post, we have talked about the concept of the surface of a prism and came to know that the surface area of the prism is the total sum of the total space filled by the flat surfaces of the prism. We have also come to know about the concept of base area and perimeter of prism. In the end we have also discussed the practical problem based on these topics for better understanding.