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Total Surface Area of Cuboid: Definitions, Derivations, How to Calculate, Difference Between TSA and CSA, and Uses

Nikita Parmar

Updated on 01st July, 2024 , 7 min read

Cuboid

Total Surface Area of Cuboid: Comprehensive Overview

A cuboid's surface area is the total space filled by all of its surfaces/sides. A cuboid is a three-dimensional form with six rectangular sides. A cuboid, also known as a regular hexahedron, is a regular hexahedron with six rectangular faces, eight vertices, and twelve edges with congruent, opposite faces. It is a three-dimensional rectangle with four lateral sides and two top and bottom faces. Bricks, geometric boxes, shoe boxes, packing boxes, and other cuboids are commonplace. 

Total Surface Area of Cuboid: What is a Cuboid?

A cuboid is a three-dimensional shape or solid with six rectangular faces. A cuboid's faces are rectangles, and all of its corners are 90 degrees. It has eight vertices and twelve edges. A cuboid's opposing faces are always equal. It signifies that the cuboid's opposed surfaces are in the same dimension. The cuboid's measurements are Total Surface Area (TSA), Lateral or Curved Surface Area (CSA), and volume. The surface areas of cubes are measured in square units, whereas the volume is measured in cubic units.

Total Surface Area of Cuboid

Total Surface Area of Cuboid: What is the Surface Area of Cuboid?

The surface area of a cuboid is the entire area covered by all of its surfaces, and because the cuboid is the 3-D form of a rectangle, the height of the cuboid is also involved in determining the surface area. In 3-D figures, surface area and volume are determined. A cuboid's surface area is measured in square units (for example, cm2, m2, etc).

Total Surface Area of Cuboid: What is Cuboid Surface Area?

The cuboid's surface area can be of two types-

  1. Total Surface Area
  2. Lateral Surface Area or Curved Surface Area

Read more about the Formula of the CSA of Cylinder.

Total Surface Area of Cuboid: Cuboid Formula?

Before delving into the idea of the area, consider the dimensions of a cuboid-

The letters l, w, and h stand for length, width, and height, respectively.

Total Surface Area of Cuboid

A cuboid's total surface area (TSA) is equal to the sum of the areas of its six rectangular sides, which is given by-

Total Cuboid Surface Area (TSA) = 2 (lw + wh + lh) square units.

The formula above calculates the total surface area of a cuboid with all six faces.

Lateral Surface Area of Cuboid

A cuboid's lateral surface area is the total of the four planes of a rectangle, except the top (upper) and bottom (lower) surfaces. The Lateral Surface Area (LSA) of a cuboid is given mathematically as-

Lateral Surface Area (LSA) = 2 (lh + wh) 

                                            = 2 h (l + w) square units.

Total Surface Area of Cuboid

Total Surface Area of Cuboid: Derivation of the Total Surface Area of Cuboid

Because the cuboid has six rectangular sides, its total surface area is determined as follows-

Assume that l, w, and h are the cuboid's length, breadth, and height, respectively.

Thus,

The area of cuboid's front face = l x h

The area of cuboid's rear face region = l x h

The area of cuboid's upper face region = l x w

The area of cuboid's bottom face region = l x w

The area of cuboid's left face region = h x w

The area of cuboid's right face region = h x w

As the total surface area of a cuboid equals the sum of all its faces, the TSA of a cuboid is-

Total Surface Area of Cuboid = lh + lh + lw+ lw+ hw+ hw

Total Surface Area of Cuboid = 2 lh + 2 lw + 2 hw

Total Surface Area of Cuboid = 2 (lh + lw+ hw) 

As a result, the cuboid's total surface area is 2 (lh + lw + hw) square units.

Total Surface Area of Cuboid: How to Calculate the Surface Area of a Cuboid?

The surface area of a cuboid is the entire area of the cuboid's sides/surfaces. The following are the steps below to figure out how to get the surface area of a cuboid-

Step 1: Examine and record the measurements of each size. If the dimensions have multiple units, combine them into one.

Step 2: Calculate the total surface area of the cuboid using the formula TSA = 2 (lb + bh + lh) square units.

Step 3: Make a note of the area included in the appropriate units.

Total Surface Area of Cuboid

Total Surface Area of Cuboid: Sample Example for the Calculation of the Total Surface Area of a Cuboid

Determine the total surface area and lateral surface area of a cuboid with dimensions of 9 inches in length, 3 inches in width, and 5 inches in height.

Solution: In this case, Length (l) = 9 inches

                                  Width (w) = 3 inches

                                  Height (h) = 5 inches.

Total Surface Area of Cuboid = 2 (lw + wh + lh) 

                                               = 2 [(9 x 3) + (3 x 5) + (5 x 9)] 

                                               = 2 [27 + 15 + 45] 

                                               = 2 x 87 

                                               = 174 in²

Lateral surface area of Cuboid = 2 h(l + w) 

                                                 = (2 x 5) (9 + 3) 

                                                 = 270 in2

Read more about the Perimeter of Cuboid and the Volume of Cuboid Formula.

Total Surface Area of Cuboid: Difference between TSA and CSA of Cuboid

The fundamental difference between TSA and CSA of the cuboid is that TSA covers all six cuboid faces, whereas CSA only includes the cuboid's vertical faces. TSA measures the overall area of the cuboid that must be covered or painted, whereas CSA measures the area of the curved surface that wraps around the cuboid. The table below details the differences between the cuboid's total surface area and curved surface area-

Parameters

TSA of Cuboid

CSA of Cuboid

Definition

The sum of the areas of all six cuboid faces. 

The sum of the areas of the cuboid's four vertical sides.

Formula 

TSA = 2 (lw + wh + lh)

CSA = 2h (l + w)

Examples

If l = 5, w = 9, h = 3, then

TSA = 2 (5 × 9 + 9 × 3 + 5 × 3) = 174 cm2.

If l = 8, w = 4, and h = 6, then

CSA = 2(6) (8 + 4) = 144 cm3.

Involves

It involves the cuboid's whole surface.

Only the cuboid's vertical faces are involved.

Read more about the Difference Between Cube and Cuboid.

Total Surface Area of Cuboid: Uses of Surface Area of Cuboid

The surface area of a cuboid has several real-world applications in engineering, building, manufacturing, architecture, and other industries. Here are a few uses of the total surface area of cuboids-

  1. Heat transmission: The surface area of a cuboid is also used to measure the rate of heat transmission in a solid object; hence, a material's surface area is an important factor in its thermal conductivity.
  2. Manufacturing: The surface area of cuboids is used in manufacturing to calculate the quantity of material required to construct a product.
  3. Packaging: Knowing the surface area of a cuboid may assist businesses in determining the quantity of material needed and ensuring that the product fits correctly inside the packaging.
  4. Painting: Knowing how much paint is needed to cover a rectangular box or cuboid is essential for painters when estimating the cost of paint for a job.

Frequently Asked Questions

What is the Cuboid Surface Area Unit?

Ans. A cuboid's surface area is measured in square units such as cm2, m2, inch2, and so on. This is due to the fact that surface area is also area.

How do you calculate a cuboid's surface area and volume?

Ans. The total surface area of a cuboid is the sum of the areas of all its faces, and the formula is Cuboid surface area = 2 (lb + bh + hl). The volume of the cuboid is lbh.

What is the distinction between total surface area and curved surface area?

Ans. The primary distinction between total surface area (TSA) and curved surface area (CSA) is that TSA refers to the area of all the faces of the solid, whereas CSA refers to the size of the solid's curved region, which excludes the areas of the top and bottom regions.

What is a cuboid's most recent surface area?

Ans. A cuboid's lateral surface area is the area of four faces other than the top and bottom. The lateral surface area may be calculated using the formula LSA = 2h(l + b).

What are the total and lateral surface areas?

Ans. The total surface area of any solid is the sum of all the faces' areas, whereas the lateral surface area is the area of walls, i.e., the faces other than the top and bottom faces.

What is the formula for the total surface area of cuboid?

A cuboid has six sides (4 lateral sides, 1 base, and 1 top), hence the sum of the areas of all six sides equals the total surface area of a cuboid. The formula for calculating the total surface area of a cuboid is TSA = 2 (lb + bh + lh) square units.

What is the CSA of a cuboid?

A cuboid's curved surface area is the total area of its faces, omitting the top and bottom. As a result, the CSA of a cuboid is 2h (l + b), where h, l, and b represent height, length, and width, respectively.

What is TSA of cube?

The TSA of the cube can be calculated by multiplying the square of its side length by 6. The surface area of a cube with side length "a" can be calculated using the formula "6a²". The total surface area of a cube is (6 × side²) square units.

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