Surface Area of Cuboid: A cuboid is a three-dimensional (3D) geometrical object which consists of 6 rectangular faces. All angles of a cuboid are right angles and faces opposite to each other are equal. The surface area is defined as the area of faces of a closed surface. A cuboid is a 3D shape with 6 faces where each face resembles a rectangle, the surface area of a cuboid will be the sum of the areas of all these six rectangular faces.

To find out the surface area of a cuboid, we need to know the sum of the areas of the six surfaces of the cuboid. The area of the surfaces except the top and bottom is called the lateral surface area of cuboid. We can easily find out the surface area of a cuboid using some simple geometric formulas. Read on to find out more about the derivation of surface area of cuboid, its applications, and examples.

What is the Total Surface Area of Cuboid?

A cuboid is a solid three-dimensional object with six faces that are rectangular in shape, eight vertices, and twelve edges.

Properties of a Cuboid

The following are some of the properties of a cuboid:

1. A cuboid has \(6\) rectangular faces
2. A cuboid has \(8\) corner points which are known as vertices
3. A cuboid has \(12\) line segments joining two vertices known as edges
4. All angles in a cuboid are right angles
5. The edges opposite to each other are parallel

Definition of Surface Area of Cuboid

The surface area of the cuboid is calculated by adding the total area of all its surfaces. A cuboid is a three-dimensional (dimensions being length, breadth, and height) solid shape. So, the main factors that affect the surface area of a cuboid are the dimensions of length, breadth, and height. The change in any of these dimensions of a cuboid changes its surface area. The metric units of surface area are square meters or square centimetres and the USCS units used to represent the surface area are, square inches or square feet.

Surface Area of Cuboid Formula

We mainly focus on two kinds of surface areas of a cuboid:

• Total surface area
• Lateral surface area

We can find the total surface area of cuboid by summing up the area of all six faces of the cuboid, whereas the lateral surface area of cuboid is found by finding the area of each face excluding the base and the top. The total surface area of cuboid is denoted by the letter “S” while the lateral surface area of cuboid is denoted by “L”. We use the length (l), breadth(b), and height of cuboid(h) to represent the total as well as lateral surface areas of a cuboid. The formulas for the same are given below:

The Total Surface Area of Cuboid, S = 2 (lb + bh + lh)
The Lateral Surface Area of Cuboid, L = 2h (l + b)

Derivation of Surface Area of Cuboid Formula

The total surface area of cuboid is the sum of areas of all \(6\) square faces of a cuboid. The faces include the top, bottom (bases), and the remaining surfaces. The lateral surface area of cuboid is the surface area of the cuboid without the top and bottom faces.

Total Surface Area of Cuboid

The total surface area of a cuboid is the sum of all \(6\) rectangular faces, including the top and bottom.

The total surface area of a cuboid is the sum of the areas of a total number of faces or surfaces that include the cuboid.

Area of face \(ABCD\) = Area of face \(EFGH = (l \times b)\)

Area of face \(CDEF\) = Area of face \(ABGH = (b \times h)\)

Area of face \(BGCF\) = Area of face \(ADHE = (l \times h)\)

Area of \((ABCD + EFGH + CDEF + ABGH + BGCF + ADHE)\)

\( = (l \times b) + (l \times b) + (b \times h) + (b \times h) + (l \times h) + (l \times h)\)

\( = 2(l \times b) + 2(b \times h) + 2(l \times h)\)

\( = 2(lb + bh + lh)\)

Lateral Surface Area of Cuboid

The lateral surface area of a cuboid is the sum of \(4\) rectangular faces, excluding the top and bottom. 

The lateral surface area of the cuboid = Area of rectangular face \(ABGH + \) Area of rectangular face \(DCEF + \) Area of rectangular face \(ADEH + \) Area of rectangular face \(BCGF\)

\( = (AB \times BG) + (DC \times CF) + (AD \times DE) + (BC \times CF)\)

\( = (b \times h) + (b \times h) + (l \times h) + (l \times h)\)

\( = 2(b \times h) + 2(l \times h)\)

\( = 2h(l + b)\)

Difference Between Surface Area of Cube and Cuboid

The cuboid is a solid three-dimensional object which has six rectangular faces. The cuboid is similar to a cube. The number of faces, vertices, and edges of a cube and cuboid is the same. But the main difference between a cube and a cuboid is the length, breadth, and height of a cuboid may vary, whereas, in a cube, the length, breadth and, height of a cube remain the same.

The lateral surface area of a cuboid \( = 2h(l + b)\) sq. units

Total surface area \( = 2(lb + bh + lh)\) sq. units

A cube is a solid three-dimensional object which has all of its six faces as squares.

The lateral surface area \( = 4 \times \) Area of one square face  \( = 4{x^2}\)
The total surface area of cube \({\rm{ = LSA + }}\) Area of top and bottom faces
\({\rm{TSA = 4}}{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{x}}^{\rm{2}}}{\rm{ = 6}}{{\rm{x}}^{\rm{2}}}\)

What Are Some of the Cuboid Examples?

The textbooks we read, the lunch box we carry to school, the mattresses on which we sleep, and the bricks we use to construct a house, etc., are well-known examples of a cuboid present in our environment.

Examples on Surface Area of Cuboid

Q.1. A plastic box \({\rm{1}}{\rm{.5m}}\) long, \({\rm{1}}{\rm{.25m}}\) wide, and \({\rm{65cm}}\) deep is to be made. It is open at the top. Ignoring the thickness of the plastic sheet, determine the area of the sheet required for making the box and the cost of the sheet for it, if a sheet measuring \({\rm{1}}{{\rm{m}}^{\rm{2}}}\) costs  \({\rm{₹20}}\).
Ans: Given \(l = 1.5\;{\rm{m}},b = 1.25\;{\rm{m}}\) and \(h = 65\;{\rm{cm}} = 0.65\;{\rm{m}}\)
Also, it is given that the box is open at the top. So, 
The area of the sheet required = LSA of the cuboid + Area of the base
We know that, LSA of the cuboid \( = 2h(l + b)\) sq.units
Therefore, LSA \( = 2 \times 0.65(1.5 + 1.25){{\rm{m}}^2}\)
\( = 3.575\;{{\rm{m}}^2}\)
Also, the area of the base of cuboid \( = l \times b\)
\( = 1.5 \times 1.25 = 1.875\;{{\rm{m}}^2}\)
Therefore, the area of the sheet required \( = 3.575 + 1.875 = 5.45\;{{\rm{m}}^2}\)
Cost of sheet \( = ₹20{\rm{per }}{{\rm{m}}^2}\)
Cost of sheet required for the box = Total area of the sheet required to make the cuboid box cost \({\rm{per }}{{\rm{m}}^2}\)
\( = 5.45 \times ₹20\)
\( = ₹109\)

Q.2. Find the total surface area (TSA) of a cuboid whose length, breadth, and height are \({\rm{6cm,4cm}}\) and \({\rm{2cm}}\) respectively.
Ans: Given \(l = 6\;{\rm{cm}},b = 4\;{\rm{cm}}\) and \(h = 2\;{\rm{cm}}\)
We know that the total surface area of a cuboid is \(2(lb + bh + lh)\)
So, TSA of the given cuboid \( = 2[(6 \times 4) + (4 \times 2) + (2 \times 6)]{\rm{c}}{{\rm{m}}^2}\)
\( = 2[24 + 8 + 12]{\rm{c}}{{\rm{m}}^2}\)
\( = 2(44){\rm{c}}{{\rm{m}}^2}\)
\( = 88\;{\rm{c}}{{\rm{m}}^2}\)
Therefore, TSA of cuboid \( = 88\;{\rm{c}}{{\rm{m}}^2}\)

Q.3. The length, breadth, and height of a room are \({\rm{5m,4m}}\), and \({\rm{3m}}\), respectively. Find the cost of whitewashing the walls of the room and the ceiling at the rate of \({\rm{₹7}}{\rm{.50per }}{{\rm{m}}^{\rm{2}}}\).
Ans: Given: Length of the room \({\rm{ = 5m}}\)
Breadth of the room \({\rm{ = 4m}}\)
Height of the room \({\rm{ = 3m}}\)
To find the amount of area to be painted, we need to find the total surface area of the room and subtract the base or floor area from it.
We know that the total surface area of a cuboid \( = 2(lb + bh + lh)\)
\( = 2[(5 \times 4) + (4 \times 3) + (5 \times 3)]{{\rm{m}}^2}\)
\( = 2[20 + 12 + 15]{{\rm{m}}^2}\)
\( = 94\;{{\rm{m}}^2}\)
Also, area of base \( = l \times b = 5 \times 4 = 20\;{{\rm{m}}^2}\)
Now, area to be whitewashed = Area of the cuboid – Area of base
\( = 94 – 20{{\rm{m}}^2}\)
\( = 74\;{{\rm{m}}^2}\)
Cost of whitewashing = Area of whitewashed surface rate per \({{\rm{m}}^2}\)
\( = 74 \times 7.5\)
\( = ₹555\)

Q.4. Find the lateral surface area of the following cuboid whose length, breadth, and height are\({\rm{4cm,4cm}}\), and \({\rm{10cm}}\) respectively.
Ans: Given \(l = 4\;{\rm{cm}},b = 4\;{\rm{cm}},\) and \(h = 10\;{\rm{cm}}\)
We know that the lateral surface area of a cuboid is \(2h(l + b)\) sq units
So, LSA of the given cuboid \( = 2 \times 10(4 + 4)\)
\( = 20(8){\rm{c}}{{\rm{m}}^2}\)
\( = 160\;{\rm{c}}{{\rm{m}}^2}\)
Therefore, LSA of the given Cuboid \( = 160\;{\rm{c}}{{\rm{m}}^2}\)

Q.5. Calculate the lateral and total surface area of a cuboid of dimensions \(10\;{\rm{cm}} \times 6\;{\rm{cm}} \times 5\;{\rm{cm}}\)
Ans: Given \(l = 10\;{\rm{cm}},b = 6\;{\rm{cm}}\) and \(h = 5\;{\rm{cm}}\)
We know that the lateral surface area of a cuboid is \(2h(l + b)\) sq units
So, LSA of the given cuboid \( = 2 \times 5(10 + 6)\)
\( = 10(16){\rm{c}}{{\rm{m}}^2}\)
\( = 160\;{\rm{c}}{{\rm{m}}^2}\)
Therefore, LSA of the given Cuboid \( = 160\;{\rm{c}}{{\rm{m}}^2}\)
We know that, Total surface area of a cuboid \( = 2(lb + bh + lh)\)
\( = 2[(10 \times 6) + (6 \times 5) + (10 \times 5)]{\rm{c}}{{\rm{m}}^2}\)
\( = 2[60 + 30 + 50]{\rm{c}}{{\rm{m}}^2}\)
\( = 2[140]{\rm{c}}{{\rm{m}}^2}\)
\( = 280\;{\rm{c}}{{\rm{m}}^2}\)

FAQs on Surface Area of Cuboid

The following are some the FAQs on the surface area of cuboid and the surface area of cuboid formula:

Q.1. What are the \(20\) types of polygons?
Ans: The \(20\) polygons based on their sides are given below:

Sides	Name
One	Monogon
Two	Bigon
Three	Triangle
Four	Tetragon or Quadrilateral
Five	Pentagon
Six	Hexagon
Seven	Heptagon
Eight	Octagon
Nine	Nonagon or Enneagon
Ten	Decagon
Eleven	Hendecagon
Twelve	Dodecagon
Thirteen	Tridecagon
Fourteen	Tetradecagon
Fifteen	Pentadecagon
Sixteen	Hexadecagon
Seventeen	Heptadecagon
Eighteen	Octadecagon
Nineteen	Enneadecagon
Twenty	Icosagon

Q.2. What is the TSA of a cuboid?
Ans: The TSA or total surface area of cuboid is the sum of all \(6\) rectangular faces, including top and bottom.
TSA of a cuboid \( = 2(lb + bh + lh)\)

Q.3. What is the surface area of cuboid formula?
Ans: The formula for finding the areas of a cuboid are:
The lateral surface area of a cuboid \( = 2h(l + b)\) sq.units
Total surface area of a cuboid \( = 2(lb + bh + lh)\) sq.units.

Q.4. What is the surface area of a \(3 \times 3 \times 3\) cube?
Ans: We know that the total surface area of a cube \( = 6{x^2}\) sq.units
Given that, \(x = 3\) units
So, TSA of a cube \( = 6 \times 3 \times 3\) sq.units \( = 54\) sq.units.

Q.5. What is the volume of a cuboid?
Ans: The volume of a cuboid \( = l \times b \times h\) cubic units.

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