Surface Area of Cuboid

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Area of Cuboid

Imagine objects like a lunch box, television set, shoebox, carton box, bricks, book, mattresses and you would know what a cuboid is and how it looks. These shapes are cuboid. Like said, a cuboid is a 3-D geometrical object which consists of 6 rectangular faces. All angles of a cuboid are right angles and faces opposite to each other are equal. A cuboid is also known as a rectangular solid or a rectangular prism. In a cuboid, the length, width and height may be of different measurements.


Is Cube a Cuboid?

Objects such as Rubik’s cube, ice, dice, Sudoku, sugar cubes, casseroles and milk crates etc are examples of another 3 dimensional shape called a cube. Factually, a cube is a unique form of cuboid in which all sides are similar and squares.


Best Way to Identify a Cuboid

In a cuboid, each face is in the form of a rectangular shape and the corners or the vertices are 90-degree angles. Also, if the opposite faces are always equal to one another then it’s a cuboid. For example, a mattress is a cuboid. It consists of 6 surfaces of which each opposite pair is of similar dimensions.


What is the Volume of Cuboids?

We can simply find the volume of a cuboid by multiplying the base area with the height. Thus,

volume of cuboid (V) = A  x h

or simply

V = l × b × h


Total Surface Area of Cuboid

If l is the length, b is the breadth and h is the height of a given cuboid, then the sum of areas of 6 rectangles of a cuboid provides the TSA of the cuboid.


Total Surface Area of Cuboid Formula

TSA of cuboid formula = 2 (lw + wh + hl)

Where,

L = length

W= width b= breadth

H = height


Lateral Surface Area Of Cuboid

The sum of the area of 4 side faces i.e. leaving the top and the bottom face provides the LSA of a cuboid. An example of the LSA is the sum of the area of the four walls of a room.


Lateral Surface Area of Cuboid Formula

LSA of cuboid formula = 2 (lh + wh) = 2 h (l + w)

Or simply, 2 (l+w)h

Where,

L = length

W= width or b= breadth

H = height


Solved Examples on Surface Area of Cuboid


Example 1:

The length, width and height of a cuboid are 11cm, 9cm and 15cm respectively. Calculate the total surface area of the cuboid.


Solution:

TSA of a cuboid is given by:  2 (l*w + w*h + w*l)

Given that:

l = 11cm

w = 9cm

h = 15cm

By substituting the values in the expression we will obtain,

TSA = 2 (11*9 + 9*15 + 15*11)

TSA = 2(99 + 135 + 165)

TSA = 2 * 399

TSA = 798cm²


Example 2:

Find out the lateral surface area of a cube having an edge of 20cm?

 

Solution:

We know that the LSA of a cuboid is given by 2(l+b)h

Now, since a cube is also a cuboid in which l=b=h=a, thus LSA of a cube = 2(a+a)

Or simply,

a = 4a2

 

Formula for Lateral Surface Area of Cube = 4a2

Given that a = 20 cm.

Therefore,

LSA = 4(202) = 1600 cm2


Example 3:

Williams built a rectangular cardboard box 20 cm high. It has a square base and a volume of 2000 cm³. Then he realized that he did not require a box that elongated, so he cut short the height of the box decreasing its volume to 1,000 cm³. Find out the height of the new box and is the new box cubicle?


Solution:

Volume of cuboid (V)  = length × width × height = Base area × height.

Given that,

V = 2000 cm³,

height = 20 cm

Substituting the values in the formula, we obtain

Base area = 2000/20 = 100 cm²

We also know that the base of this box is a square.

Thus, it indicates that the length = width.

Hence, the length of square base = √100 = 10 cm

After shortening of height, new volume = 1000

= 10 ×10 x new height

Thus, new height = 1000/ 10 × 10 = 10 cm

As all the dimensions of the solid, l, w, h measure similar, the resulting solid is also a cube.

FAQ (Frequently Asked Questions)

Q1. What is the Diagonal of a Cuboid?

Answer: The length of the longest diagonal of a cuboid is given as:

Length of diagonal of cuboid = √ (l² + b² + h²), where l= length, b= breadth, h= height

Q2. What are the Properties of a Cube Number?

Answer:

  • Cubes of positive numbers are positive invariably. 

             For example, cube of +3 is = (+3) × (+3) × (+3) = +27

  • Cubes of positive numbers are invariably negative. 

            For example, cube of -3 is = (-3) × (-3) × (-3) = -27

  • Cubes of even numbers even invariably.

  • Cubes of odd numbers are odd invariably.

Q3. How Do We Measure the Volume of Water?

Answer: Generally, it is not feasible to measure the volume of water until it is stored in a container, which can be a cube, cuboid, cone, cylinder, and cone etc. And once it is inside a container we need to compute the volume of the container in order to ascertain the volume of water.

Q4. What is Meant by Nets of a Cuboid?

Answer: Another way to have a perception of the surface area of a cuboid is to take into account a net of the cuboid. The net is a 2-D geometrical shape that can be molded to create a 3-D object.


Imagine cutting along some edges of a cuboid and opening it up to create a plane figure. The plane figure is what we call the net of the cuboid.

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