Surface Area of Cuboid
The surface area of a cuboid is the total space occupied by it. A cuboid is a six-faced three-dimensional shape in which each face is in the shape of a rectangle. If l, w, and h are the length, width, and the height of a cuboid, then the surface area of a cuboid formula is 2 lw + 2 wh + 2hl.
This formula can be derived by observing that a cuboid has three pairs of faces, each pair having the same area (because opposite faces are congruent), and adding up the areas of all six faces. Let us learn more about the formula of the surface area of the cuboid and solve problems based on it.
What is the Surface Area of Cuboid?
The surface area of a cuboid is the total area of all its 6 faces. Since a cuboid is a three-dimensional solid shape, the value of its surface area depends on the dimensions of its length, width, and height. The change in any of the dimensions of a cuboid changes the value of its surface area. The surface area of a cuboid is expressed in square units.
The surface area of a cuboid is an important metric as it is useful in a wide range of applications such as construction, manufacturing, and packaging. For example, when determining how much paint is needed to cover a cuboidal box, or when calculating the amount of material required to manufacture a cuboidal object, the surface area of the cuboid is an essential measurement.
Surface Area of Cuboid Formula
The formula for the surface area of a cuboid depends on the type of surface area that has been asked for. A cuboid has two kinds of surface areas:
- TSA of Cuboid : Total Surface Area (also known just as "surface area")
- LSA (or) CSA of Cuboid: Lateral Surface Area (or) Curved Surface Area
The total surface area of the cuboid is obtained by adding the area of all the 6 faces while the lateral surface area of the cuboid is calculated by finding the area of each face excluding the base and the top. The total surface area and the lateral surface area of a cuboid can be expressed in terms of length (l), width (w), and height of cuboid (h) as:
Total Surface Area of Cuboid = 2 (lw + wh + lh)
Lateral Surface Area of Cuboid = 2h (l + w)
If the surface area is given or asked without any specifications, it means that it refers to the total surface area. Let us see how to derive the formulas for the surface area of the cuboid by opening up a cuboidal box.
Total Surface Area of Cuboid (TSA)
The TSA of cuboid is sometimes just referred to as surface area of cuboid itself. A cuboid has 6 rectangular faces. Now, in order to find the total surface area of a cuboid, we need to add the area of all the 6 rectangular faces. Since each face of a cuboid is a rectangle, hence the area of the rectangle for each face is calculated and added to get the total surface area of the cuboid. Let us understand this with the help of the figure given below.
The faces of cuboid are numbered as 1, 2, 3, 4, 5, and 6 as shown in the figure given above. In other words, if we see the cuboid in the form of a two-dimensional figure as a net, we get this figure.
- Area of Rectangles 1 and 2: the area of the rectangles of the top and bottom faces = lw + lw = 2lw
- Area of Rectangles 3 and 4: the area of the rectangles of the left and right sides = wh + wh = 2wh
- Area of Rectangles 5 and 6: area of the rectangles on the front and back faces = lh + lh = 2lh
Hence, the total surface area (TSA) of the six faces = 2lw + 2wh + 2lh.
Thus, the TSA of a cuboid of dimensions l, w, and h is 2 (lw + wh + lh).
Lateral Surface Area of Cuboid (LSA)
The lateral surface area (LSA) of a cuboid is also known as the curved surface area of cuboid (CSA). This is the combined surface area of the four lateral (side) faces. In the figure given above, if we remove the top and bottom faces, we will get the area of the lateral surface area of the cuboid. The LSA (or) CSA of cuboid is,
Lateral Surface Area = Area of the four side faces
= 2lh + 2wh
= 2h(l + w)
It is important to note that CSA of cuboid only includes the vertical faces and does not include the top and bottom faces, so it is always less than the total surface area of the cuboid.
To understand this with the help of a real-life example let us imagine a room in the shape of a cuboid. The total surface area will be the combined area of the six faces of the room (the four vertical walls + the floor + the ceiling) while the lateral surface area will be the combined surface area of the four vertical walls (the areas of the floor and the ceiling are not added).
Difference Between TSA and CSA of Cuboid
The main difference between TSA and CSA of the cuboid is that TSA includes all six faces of the cuboid, whereas CSA only includes the vertical faces of the cuboid. TSA is a measure of the total area that needs to be covered or painted on the cuboid, while CSA is a measure of the area of the curved surface that wraps around the cuboid. The following table summarizes the differences between the total surface area and curved surface area of cuboid in detail.
| Property | TSA of Cuboid | CSA of Cuboid |
|---|---|---|
| Definition | Sum of areas of all 6 faces of cuboid | Sum of areas of all 4 vertical faces of cuboid |
| Formula | TSA = 2(lw + wh + lh) | CSA = 2h(l + w) |
| Involves | All faces of the cuboid | Only vertical faces of the cuboid |
| Example | If l = 4, w = 3, h = 2, then TSA = 2(4 × 3 + 3 × 2 + 4 × 2) = 52 cm2 |
If l = 4, w = 3, h = 2, then CSA = 2(2) (4 + 3) = 28 cm3 |
How to Find the Surface Area of Cuboid?
The surface area of a cuboid is the total area of each surface of a cuboid. Let us use the following steps to calculate the total surface area of a cuboid.
- Step 1: Check if the given dimensions of cuboids are in the same units or not. If not, convert the dimensions into the same units.
- Step 2: Use the formula for total surface area = 2(lw + wh + lh) if we need TSA and lateral surface area = 2h(l + w) if we need LSA.
- Step 3: Substitute the given values to get the area and express it in square units.
Example: Find the total surface area and the lateral surface area of a cuboid whose length = 6 inches, width = 4 inches, and height = 3 inches.
Solution: Here, length (l) = 6 inches, width (w) = 4 inches and height (h) = 3 inches
- Total surface area of cuboid = 2 (lw + wh + lh) = 2 [(6 × 4) + (4 × 3) + (6 × 3)] = 108 in2
- Lateral surface area of cuboid = 2h(l + w) = (2 × 3)(6 + 4) = 60 in2
Applications of Surface Area of Cuboid
The surface area of a cuboid has various real-world applications in various fields such as engineering, construction, manufacturing, architecture, etc. Here are a few applications.
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Packaging: Knowing the surface area of a cuboid can help companies determine the amount of material required and ensure that the product fits properly inside the packaging.
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Painting: The information about the amount of paint needed to cover a rectangular box or cuboid is crucial for painters to estimate the cost of paint required for a project.
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Manufacturing: In manufacturing, the surface area of cuboids is used to calculate the amount of material needed to manufacture a product.
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Heat Transfer: The surface area of a cuboid is also used to calculate the rate of heat transfer in a solid object and hence surface area of a material is a critical factor in its thermal conductivity.
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FAQs on Surface Area of Cuboid
What Does the Total Surface Area of Cuboid Mean?
The total surface area of a cuboid is the sum of all its surfaces. In order to find the total surface area of a cuboid, we need to add the area of all the 6 rectangular faces. The formula for the total surface area of a cuboid is 2 (lw + wh + lh) where l = length, w = width, and h = height of the cuboid.
What is the Lateral Surface Area of a Cuboid?
The lateral surface area of a cuboid is the value of the surface area of a cuboid excluding its top and bottom surfaces. The formula for the lateral surface area of a cuboid is expressed as, 2h(l + w) where l = length, w = width, and h = height of the cuboid.
Where to Find the Surface Area of Cuboid Calculator?
The area of cuboid can be calculated using the calculator. It shows the surface area just by entering the values of length, width, and height of the cuboid.
What is the Difference Between the TSA and the LSA of a Cuboid?
The difference between total surface area and the lateral surface area of a cuboid is given below:
- The total surface area of a cuboid is the sum of the areas of all 6 faces and is calculated by the formula 2(lw + wh + lh).
- The lateral surface area of a cuboid is the sum of the areas of faces excluding the top and the base and is calculated by the formula 2h(l + w)
What is the Total Surface Area of a Cuboidal Box Without Lid?
The total surface area of a cuboidal box without a lid can be given in two ways:
- Method 1: Total surface area of a cuboidal box without lid = Total surface area of the cuboid - 1 rectangular face;
- Method 2: Total surface area of a cuboidal box without lid = Lateral surface area of cuboid + 1 rectangular face;
It should be noted that both the methods result in the same answer.
What is the Unit for Surface Area of Cuboid?
The surface area of a cuboid is expressed in square units like cm2, m2, inch2, and so on. This is because surface area is also area.
What is the TSA and CSA of a Cuboid?
The TSA and CSA of a cuboid are the abbreviations of the Total Surface Area (TSA) and Curved Surface Area (CSA) of a cuboid. The total surface area is the sum of the surface area of all 6 faces while curved surface area is the measure of surface area excluding the top and bottom faces.
What is the Total Surface Area of a Cuboid Formula?
The formula that is used to find the total surface area of a cuboid is 2 (lw + wh + lh); where l = length, w = width, and h = height of the cuboid.
What is the Lateral Surface Area of a Cuboid Formula?
The formula that is used to find the lateral surface area of a cuboid is 2h(l + w) where l, w, and h are the length, width, and height of the cuboid respectively.