Surface Area of a Prism: In mathematics, a prism is an essential member of the polyhedron family and is defined as a three-dimensional shape having two identical polygons facing each other that are connected by rectangular or parallelogram faces laterally. The identical polygons can be triangles, squares, rectangles, pentagons, or any other n-sided polygon and are called the bases of the prism. The other faces of a prism are parallelograms or rectangles.
In this article, we will discuss different types of prisms, and the surface area of prism formula, with examples and practice problems.
What is the Surface Area of Prism?
The surface area of a prism is referred to as the total area enclosed by all its faces. To determine a prism’s surface area, we must calculate the areas of each of its faces, and then add the resulting areas. A prism has two kinds of surface areas, namely the lateral surface area and the total surface area. Â The area occupied by a prism’s faces, excluding the two parallel faces (bases of a prism), is referred to as its lateral surface area.
The lateral surface area of a prism = [Base perimeter × height] square units
Now, a prism’s total surface area is the sum of the areas of its two bases and its lateral surface area.
The general formula for calculating the total surface area of any type of right prism is:
The total surface area of a Prism = [2 (Base Area) + (Base perimeter × height)] square units
Different Types of Prism
There are different types of prisms based on the shape of the base of a prism, such asÂ
- Triangular prisms,
- Square prisms,Â
- Rectangular prisms,Â
- Pentagonal prisms,Â
- Hexagonal prisms,Â
- Octagonal prisms.
Triangular Prism
A prism with a triangular base is referred to as a triangular prism. A triangular prism consists of three inclined rectangular surfaces and two parallel triangle bases. Let “H” be the height of the triangular prism; “a, b, and c” are the sides’ lengths, and “h” is the height of the triangular bases.Â
The perimeter of a triangular base (P) = Sum of its three sides = a + b + c
The area of a triangular base (A) = ½ × base × height =  ½ bh
We know that the general formula for the lateral surface area of a right prism is L. S. A. = PH, where P is the base perimeter, and A is the base area.
By substituting all the values in the general formula we get,
The lateral surface area of a triangular prism = (a + b +c)H square units
where,
a, b, c are sides of triangular base
H is height of  triangular prism
We know that the general formula for the total surface area of a right prism is T. S. A. = PH+2A, where P is the base perimeter, A is the base area, and H is the height of the prism.
By substituting all the values in the general formula we get
The total surface area of the triangular prism = (a + b + c)H + 2 × (½ bh)
The total surface area of the triangular prism = (a + b + c)H + bh square units
where,
a, b, c are sides of triangular base
H is height of  triangular prism
h is height of triangle
Rectangular Prism
A prism with a rectangular base is referred to as a rectangular prism. A rectangular prism consists of four rectangular surfaces and two parallel rectangular bases. Let the prism’s height be “h” and its rectangular bases’ length and width be “l” and “w,” respectively.
The perimeter of a rectangular base (P) = Sum of its four sides = 2 (l + w)
The area of a rectangular base (A) = length × width =  l × w
We know that the general formula for the lateral surface area of a right prism is L. S. A. = PH, where P is the base perimeter, and A is the base area.
By substituting all the values in the general formula we get,
The lateral surface area of a rectangular prism = 2h(l + w) square units
where,
l is length
w is width
h is height
We know that the general formula for the total surface area of a right prism is T. S. A. = PH+2A, where P is the base perimeter, A is the base area, and H is the height of the prism.
By substituting all the values in the general formula we get
The total surface area of the rectangular prism = 2h(l + w) + 2(l × w)
= 2 lh + 2 wh + 2 lw
The total surface area of the rectangular prism = 2 (lh + wh + lw) square units
where,
l is length
w is width
h is height
Square Prism
A prism with a square base is referred to as a square prism. A square prism consists of four rectangular surfaces and two parallel square bases. Let the prism’s height be “h” and its square bases’ lengths be “s”.
The perimeter of a square base (P) = Sum of its four sides = s + s + s + s = 4s
The area of a square base (A) = (length of the side)2 = Â s2
We know that the general formula for the lateral surface area of a right prism is L. S. A. = PH, where P is the base perimeter, and A is the base area.
By substituting all the values in the general formula we get,
The lateral surface area of a square prism = 4sh square units
where,
s is side of square base
h is height of square prism
We know that the general formula for the total surface area of a right prism is T. S. A. = PH+2A, where P is the base perimeter, A is the base area, and H is the height of the prism.
By substituting all the values in the general formula we get
The total surface area of the square prism = [4sh + 2s2] square units
where,
s is side of square base
h is height of square prism
Pentagonal Prism
A prism with a pentagonal base is referred to as a pentagonal prism. A pentagonal prism consists of five inclined rectangular surfaces and two parallel pentagonal bases. Let “h” be the height of the pentagonal prism; “a and b” be the apothem length and side lengths of the pentagonal bases.Â
The perimeter of a pentagon base (P) = Sum of its five sides = 5b
The area of a pentagon base (A) = 5/2 x (apothem length) x (length of the side) = 5ab
We know that the general formula for the lateral surface area of a right prism is L. S. A. = PH, where P is the base perimeter, and A is the base area.
By substituting all the values in the general formula we get,
The lateral surface area of a pentagonal prism = 5bh square units
where,
b is side of pentagonal base
h is the height of pentagonal prism
We know that the general formula for the total surface area of a right prism is T. S. A. = PH+2A, where P is the base perimeter, A is the base area, and H is the height of the prism.
By substituting all the values in the general formula we get,
The total surface area of the pentagonal prism = [5bh + 5ab] square units
where,
b is side of pentagonal base
a is apothem length.
h is the height of pentagonal prism
Hexagonal Prism
A prism with a hexagonal base is referred to as a hexagonal prism. A hexagonal prism consists of six inclined rectangular surfaces and two parallel hexagonal bases. Let “h” be the height of the hexagonal prism; “a” be the side lengths of the hexagonal bases.Â
The perimeter of a hexagon base (P) = Sum of its six sides = 6a
The area of a hexagon base (A) = 6 x (Area of an equilateral triangle)Â
A = 6 x (√3a2/4) ⇒  A = 3√3a2/2
We know that the general formula for the lateral surface area of a right prism is L. S. A. = PH, where P is the base perimeter, and A is the base area.
By substituting all the values in the general formula we get,
The lateral surface area of a hexagonal prism = 6ah square units
where,
a is side of hexagonal base
h is height of hexagonal base
We know that the general formula for the total surface area of a right prism is T. S. A. = PH+2A, where P is the base perimeter, A is the base area, and H is the height of the prism.
By substituting all the values in the general formula we get
The total surface area of the hexagonal prism =  [6ah +3√3a2 ]  square units
where,
a is side of hexagonal base
h is height of hexagonal base:
The table given below provides the formula for different types of prisms:
Shape
|
Base of the prism
|
Lateral surface area
 [Base perimeter × height]
|
Total Surface Area
[(2 × Base Area) + (Base perimeter × height)]
|
Triangular Prism
|
Triangle
|
(a + b +c)H square units
|
(a + b + c)H + bh square units
|
Rectangular Prism
|
Rectangle
|
2h(l + w) square units
|
2 (lh + wh + lw) square units
|
Square Prism
|
Square
|
 4sh square units
|
[4sh + 2s2] square units
|
Pentagonal Prism
|
Pentagon
|
5bh square units
|
[5ab + 5bh] square units
|
Hexagonal PrismÂ
|
Hexagon
|
6ah square units
|
[3√3a2 + 6ah] square units
|
Surface Area of a Prism Solved Examples
Problem 1: What is the height of a prism whose base area is 36 square units, its base perimeter is 24 units, and its total surface area is 320 square units?
Solution:
Given data,
Base area = 36 square units
Base perimeter = 24 units
The total surface area of the prism = 320 square units
We have,
The total surface area of the prism = (2 × Base Area) + (Base perimeter × height)
⇒ 320 = (2 × 36)+ (24 × h)
⇒ 24h = 248 ⇒ h = 10.34 units
Hence, the height of the given prism is 10.34 units.
Problem 2: Find the total surface area of a square prism if the height of the prism and the length of the side of the square base are 13 cm and 4 cm, respectively.
Solution:
Given data,
The height of the square prism (h) = 13 cm
The length of the side of the square base (a) = 4 cm
We know that,
The total surface area of a square prism = 2a2 + 4ah
 = 2 × (4)2 + 4 × 4 × 13
= 32 + 208 = 240 cm2
 Hence, the total surface area of the given prism is 240 sq. cm.
Problem 3: Determine the base length of a pentagonal prism if its total area is 100 square units and its height and apothem length are 8 units and 5 units, respectively.
Solution:
Given data,
The total surface area of the pentagonal prism = 100 square units
The height of the prism (h) = 8 units
Apothem length (a) = 5 units
We know that,
The total surface area of the pentagonal prism = 5ab + 5bh
⇒ 100 = 5b (a+ h)
⇒ 100/5 = b (5 + 8)
⇒ 20 = b × (13) ⇒ b = 25/16 = 1.54 units
Hence, the base length is 1.54 units
Problem 4: Determine the height of the rectangular prism and the total area of a rectangular prism if its lateral surface area is 540 sq. cm and the length and breadth of the base are 13 cm and 7 cm, respectively.
Solution:
Given data,
The length of the rectangular base (l) = 13 cm
The width of the rectangular base (w) = 7 cm
The lateral surface area of the prism = 540 sq. cm
We have,
The lateral surface area of the prism = Base perimeter × height
⇒ 540 = 2 (l + w) hÂ
⇒ 2 (13 + 7) h = 540
⇒ 2 (20) h = 540 ⇒ h = 13.5 cm
We know that,
The total surface area of the rectangular prism = 2 (lw + wh + lh)
= 2 × (13 × 7 + 7 × (13.5) + 13 × (13.5))
= 2 × (91 + 94.5 + 175.5) = 722 sq. cm
Hence, the height and total surface area of the given rectangular prism are 13.5 cm and 722 sq. cm, respectively.
Problem 5: Determine the surface area of the regular hexagonal prism if the height of the prism is 12 in and the length of the side of the base is 5 in.
Solution:
Given data,
The height of the prism (h) = 12 in
The length of the side of the base (a) = 6 in
The surface area of a regular hexagonal prism = 6ah + 3√3a2
= 6 × 5 × 12 + 3√3(5)2
= 360 + 75√3
= 360 + 75 × (1.732) = 489.9 sq. in
Hence, the surface area of the given prism is 489.9 sq. in.
Problem 6: Calculate the lateral and total surface areas of a triangular prism whose base perimeter is 25 inches, the base length and height of the triangle are 9 inches and 10 inches, and the height of the prism is 14 inches.
Solution:
Given data,
The height of the prism (H) = 14 inches
The base perimeter of the prism (P) = 25 inches
The base length of the triangle = 9 inches
The height of the triangle = 10 inches
We know that,
The lateral surface area of the prism = Base perimeter × height
= 25 × 14= 350 sq. in
Area of the triangular base (A) = ½ × base × height = 1/2 × 9 × 10 = 45 sq. in
The total surface area of the triangular prism = 2A + PHÂ
= 2 × 45 + 25 × 14 = 90 + 350 = 440 sq. inÂ
Hence, the prism’s lateral and total surface areas are 350 sq. in and 440 sq. in, respectively.
Practice Problems on Surface Area of a Prism
1. Given a rectangular prism with dimensions:
- Length = 6 cm
- Width = 4 cm
- Height = 5 cm
Calculate the total surface area.
2. Consider a triangular prism with dimensions:
- Base of the triangle = 8 cm
- Height of the triangle = 6 cm
- Length of the prism = 10 cm
Find the total surface area.
3. Determine the surface area of a regular pentagonal prism with:
- Side length of the base = 7 cm
- Height of the prism = 9 cm.
4. Calculate the surface area of a hexagonal prism with:
- Side length of the regular hexagonal base = 10 cm
- Height of the prism = 12 cm.
FAQs on Surface Area of a Prism
What is a Prism in Geometry?
A prism is a three-dimensional shape with two congruent parallel bases and rectangular or parallelogram lateral faces connecting them. Prisms come in various forms, such as rectangular prisms, triangular prisms, and pentagonal prisms, each with unique characteristics.
How Do You Find the Surface Area of a Prism?
To find the surface area of a prism, calculate the areas of all its faces and then sum them up. For a rectangular prism, the surface area formula is 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height. For other types of prisms, such as triangular or pentagonal prisms, additional formulas for base area and lateral area may be needed.
What Are the Properties of a Prism?
Prisms have several key properties:
- They have two congruent parallel bases.
- The lateral faces are all parallelograms.
- The altitude (height) is the perpendicular distance between the two bases.
- The bases are identical in shape and size.
- The cross-section parallel to the bases is always the same shape and size as the bases.
What Are Some Real-Life Examples of Prisms?
Prisms can be found in various everyday objects and structures. Examples include:
- Rectangular prisms: Buildings, cereal boxes, books.
- Triangular prisms: Roofs of houses, wedge-shaped objects.
- Pentagonal prisms: Some types of columns, certain architectural structures.
- Hexagonal prisms: Certain types of crystals, some packaging containers.
Why is Surface Area Important in Prisms?
Surface area is crucial in prisms as it represents the total area of all the surfaces (faces) of the prism. Understanding the surface area helps in various practical applications, such as calculating the amount of material needed to build or cover a prism-shaped object, determining heat transfer rates, and optimizing packaging design.
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