Cuboid is a solid shape in geometry. It is a hexahedron with quadrilaterals as its faces. It has 8 vertices, 12 edges and 6 faces. The surface area of a cuboid is the total space occupied by all its surfaces and sides. In geometry, a three-dimensional shape having 6 rectangular faces is called a cuboid. A cuboid is also known as a regular hexahedron and has six rectangular faces, eight vertices and twelve edges with congruent opposite faces. Cuboid is a three-dimensional form of a rectangle with four lateral faces and two faces at the top and bottom. Some examples of cuboid in daily life are bricks, lunch boxes, shoe boxes, refrigerators, etc.

What is Surface Area of Cuboid?

The surface area of a cuboid is the total area of all its outer faces. Since a cuboid is a 3D shape like a stretched rectangle, you need to know its length, width, and height to calculate the surface area. The surface area tells you how much space the outside of the cuboid covers. We measure surface area in square units like square centimeters (cm²), square meters (m²), or square inches (in²). Some common examples of cuboids you see every day are textbooks, lunch boxes, mattresses, and bricks. 

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A cuboid has 2 types of surface area. The surface area of a cuboid is calculated based on what type of surface is required. The formulas for both types of surface areas are different. The 2 types of surface area of cuboid are:

1. Total surface area of cuboid
2. Lateral surface area of cuboid

Total Surface Area of Cuboid

Total surface area of cuboid is the area occupied by all its 6 rectangular faces. The total surface area of a cuboid can be calculated by first calculating the area of all the faces and then adding the area of all \(6\) faces. Total surface area of a cuboid is given by \(TSA = 2(lb + bh + hl)\), where \(l\), \(b\) and \(h\) are length, breadth and height of a cuboid.

Lateral Surface Area of Cuboid

Lateral surface area of cuboid is the area occupied by 4 rectangular faces, excluding the top and bottom. The lateral surface area of a cuboid is also known as the curved surface area of a cuboid. The surface area of a cuboid can be calculated by first calculating the area of the 4 rectangular faces and then adding the area of all \(4\) rectangular faces.

Lateral or curved surface area of a cuboid is given by \(LSA = CSA = 2h(l + b)\), where \(l\), \(b\) and \(h\) are length, breadth and height of a cuboid.

Surface Area of Cuboid Formula

Total and lateral surface areas of a cuboid are the two types of surface areas of a cuboid. The total surface area of a cuboid is the sum of all 6 rectangular faces whereas lateral or curved surface area of a cuboid is the sum of 4 rectangular faces, excluding the top and bottom faces. The formula for both the total surface and lateral surface areas of a cuboid are different.

Derivation of Formula for Total Surface Area of Cuboid

To find the total surface area of a cuboid, you first calculate the area of each of its six faces. Then, you add all these areas together. Here, l is the length, b is the width (breadth), and h is the height of the cuboid.

We know that in a cuboid, opposite faces are equal. Then, we have

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

Area of face ABGF = Area of face CDEH = \(b \times h\)

Area of face ADEF = Area of face BCHG = \(h \times l\)

Now, the total surface area of a cuboid = Area of \([ABCD+EFGH+ABGF+CDEH+ADEF+BCHG]\)

\(\Rightarrow\) Total surface area of a cuboid = \(2 \times \) Area of \( [ABCD+ABGF+ADEF]\)

\(\Rightarrow\) Total surface area of a cuboid = \(2 \times (l \times b + b \times h + h \times l)\)

Therefore, the total surface area of a cuboid, \(TSA = 2(lb + bh + hl)\).

Derivation of Formula for Lateral Surface Area of Cuboid

Lateral surface area of a cuboid is the area occupied by its 4 rectangular faces, excluding the top and bottom faces of the cuboid.

Hence, the lateral surface area of a cuboid = Total surface area of a cuboid – the area of top and bottom faces of the cuboid.

\(\Rightarrow\) Lateral surface area of a cuboid = \(2(lb + bh + hl)\) – Area of \([ABCD + EFGH]\)

Since, we know that

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

\(\Rightarrow\) Lateral surface area of a cuboid = \(2(lb + bh + hl) – 2 \times \) Area of \([ABCD]\)

\(\Rightarrow\) Lateral surface area of a cuboid = \(2(lb + bh + hl) – 2 \times (lb)\)

\(\Rightarrow\) Lateral surface area of a cuboid = \(2(bh + hl)\)

\(\Rightarrow\) Lateral surface area of a cuboid = \(2h(b + l)\)

Therefore, the lateral or curved surface area of a cuboid, \(LSA = CSA = 2h(b + l)\).

Learn about Area of a Quadrilateral

How to Find Surface Area of Cuboid

The surface area of a cuboid is the total area of each surface of a cuboid. The steps to calculate the total surface area of a cuboid are given below:

The surface area of a cuboid is the total area covered by all its sides. Here’s how you can calculate it:

Step 1: Measure the length, width, and height of the cuboid. Make sure all measurements are in the same unit.

Step 2: Use the formula to find the total surface area of the cuboid.

Step 3: Put the numbers you measured into the formula and calculate the answer. The result will tell you the surface area in square units, like square centimeters or square meters.

Difference between Total Surface Area and Lateral Surface Area of a cuboid:

Applications of Surface Area of Cuboid

The surface area of a cuboid is useful in many real-life situations across different fields like engineering, building, manufacturing, and design. Here are some examples that show why knowing the surface area is important:

1. Packaging: Companies use the surface area of a cuboid to figure out how much material is needed to make boxes or containers. This helps them save material and ensures the product fits perfectly inside the box.

2. Painting: When painters need to paint a box-shaped object, they calculate the surface area to estimate how much paint will be required. This helps avoid wasting paint and keeps the project cost-effective.

3. Manufacturing: Manufacturers use the surface area to know how much raw material is needed to create a product, such as a metal or wooden box. This helps control costs and reduce waste.

4. Heat Transfer: Surface area plays a big role in how heat moves through objects. Knowing the surface area helps engineers design objects that either keep heat in or let heat out efficiently.

5. Wrapping Gifts: When wrapping presents shaped like boxes, knowing the surface area helps determine how much wrapping paper is needed, avoiding excess or shortage.

6. Cleaning: In industries, when cleaning box-shaped containers, the surface area helps estimate the amount of cleaning solution required to cover the entire surface properly.

7. Advertising: For products like boxes or crates, the surface area helps designers plan the space available for printing logos or labels.

Properties of Surface Area of a Cuboid

1. 1. Definition:
      A cuboid is a three-dimensional solid shape with six rectangular faces.

• Types of Surface Area:

1. • Lateral Surface Area (LSA): The area of the four vertical faces.
     Formula: LSA = 2h(l + b)

   • Total Surface Area (TSA): The area of all six faces (including top and bottom).
     Formula: TSA = 2(lb + bh + hl)

2. Units:
   Surface area is always expressed in square units (e.g., cm², m²).

3. Dependence on Dimensions:
   The surface area of a cuboid depends on its length (l), breadth (b), and height (h).

4. Symmetry of Opposite Faces:
   Opposite faces of a cuboid are equal in area.

5. Real-life Examples:
   Boxes, bricks, books, and cupboards follow the surface area properties of a cuboid.

6. Increase in Dimension Increases Area:
   Any increase in length, breadth, or height will result in a larger surface area.

Examples on Surface Area of Cuboid

Example 1: The length, breadth and height of a cuboid are \(10 cm\), \(8 cm\) and \(7 cm\) respectively. Find the lateral surface area of a cuboid.

Solution: Lateral surface area of a cuboid is given by

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

Given that:

\(l = length = 10 cm\),

\(b = breadth = 8 cm\), and

\(h = height = 7 cm\).

Substitute these values in lateral surface area of a cuboid formula,

\(LSA = 2 \times 7(10 + 8)\)

\(LSA = 14 \times 18\)

\(LSA = 252 cm^{2}\).

Example 2: The length, breadth and height of a cuboid are \(16 cm\), \(14 cm\) and \(10 cm\) respectively. Find the total surface area of the cuboid.

Solution: Total surface area of a cuboid is given by

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

Given that:

\(l = length = 16 cm\),

\(b = breadth = 14 cm\), and

\(h = height = 10 cm\).

Substitute these values in lateral surface area of a cuboid formula,

\(TSA = 2(16 \times 14 + 14 \times 10 + 10 \times 16)\)

\(TSA = 2(224 + 140 + 160)\)

\(TSA = 2 \times (524)\)

\(TSA = 1048 cm^{2}\)

Example 3: Calculate the lateral and total surface area of a cuboid with dimensions \(10 cm \times 6 cm \times 5 cm\).

Solution: Given that, \(l = length = 10 cm\), \(b = breadth = 6 cm\), and

\(h = height = 5 cm\).

(i). Lateral surface area of a cuboid, \(LSA = 2h(l + b)\)

\(LSA = 2 \times 5(10 + 6)\)

\(LSA = 10 \times 16\)

\(LSA = 160 cm^{2}\).

(ii). Total surface area of a cuboid, \(TSA = 2(l \times b + b \times h + h \times l)\)

\(TSA = 2(10 \times 6 + 6 \times 5 + 5 \times 10)\)

\(TSA = 2(60 + 30 + 50)\)

\(TSA = 2 \times (140)\)

\(TSA = 280 cm^{2}\).

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