Surface Area Of Cuboid & Cube: Formula, Derivation, Volume & Examples

In this blog, we will introduce the surface area of cuboid and cubes. We will begin with the surface area of the cuboid. Let us take a cuboid. We can see in each face of the cuboid there is a rectangle. So let us find out how many rectangles does a cuboid has? 1, 2, 3, 4, 5, and 6. So there are 6 rectangles as there are 6 faces of a cuboid. Now open this cuboid and mark its dimensions.

On opening, we have this. So we have used six rectangular pieces to cover the complete outer surface of the cuboid. This shows us that the outer surface of a cuboid is made up of six rectangles (in fact, rectangular regions, called the faces of the cuboid), whose areas can be found by multiplying the length by breadth for each of them separately and then adding the six areas together.

So, the sum of the areas of the six rectangles is:

Area of rectangle 1= (l × h) + Area of rectangle 2= (l × b)
Area of rectangle 3= (l × h)
Area of rectangle 4= (l × b)
Area of rectangle 5= (b × h)
Area of rectangle 6 = (b × h)

That is = 2(l × b) + 2(b × h) + 2(l × h) taking 2 common we have = 2(lb + bh + hl)
This gives us:

Surface Area of a Cuboid = 2(lb + bh + hl)

What about a cube? Let us see in the case of the cube. For this also open up a cube. A cuboid, whose length, breadth, and height are all equal, is called a cube. If each edge of the cube is a, then the surface area of this cube would be
2(a × a + a × a + a × a) i.e., 6a2
The surface area of a cuboid (or a cube) is sometimes also referred to as the total surface area.

Surface area of cuboid

Solved Examples

1. Mary wants to decorate her Christmas tree. She wants to place the tree on a wooden box covered with colored paper with a picture of Santa Claus on it. She must know the exact quantity of paper to buy for this purpose. If the box has length, breadth, and height as 80 cm, 40 cm, and 20 cm respectively how many square sheets of paper on the side 40 cm would she require?

Solution: Since Mary wants to paste the paper on the outer surface of the box; the quantity of paper required would be equal to the surface area of the box which is of the shape of a cuboid. The dimensions of the box are:

Length =80 cm, Breadth = 40 cm, Height = 20 cm. so we need to go for surface area of box so the surface area of the box = 2(lb + bh + hl)
= 2[(80 × 40) + (40 × 20) + (20 × 80)] cm2

= 2[3200 + 800 + 1600] cm2
= 2 × 5600 cm2 = 11200 cm2

Now let us go for the area of the sheet. The area of each sheet of the paper = 40 × 40 cm2= 1600 cm2

Therefore, the number of sheets required = surface area of box divided by area of the sheet. That is 11200 upon 1600 gives 7. Number of sheets required = 7 sheets

2. Hameed has built a cubical water tank with a lid for his house, with each outer edge 1.5 m long. He gets the outer surface of the tank excluding the base, covered with square tiles of the side 25 cm. Find how much he would spend for the tiles if the cost of the tiles is Rs 360 per dozen.

Solution: Since Hameed is getting the five outer faces of the tank covered with tiles, he would need to know the surface area of the tank, to decide on the number of tiles required.
Edge of the cubical tank = 1.5 m = 150 cm (= a)
So, surface area of the tank = 5 × 150 × 150 cm2
Area of each square tile = side × side = 25 × 25 cm2
So, the number of tiles required = surface area of tank upon area of tile. That is 5 x 150 x 150 upon 25 x 25. Cost of one dozen tiles i.e. cost of 12 tiles = Rs 360
Cost of one tile is = 360/12 =Rs30
So the cost of 180 tiles is = 180 x 30 = Rs 5400

Curved Surface Area Of Cuboid Formula:

The curved surface area of a cuboid is a total of four rectangle planes, except the top (upper) and bottom (lower) surfaces. Mathematically, the lateral or curved surface of cuboid formula is given as:

Curved surface area of cuboid formula = 2 (lh + wh) = 2 h (l + w) square units.

Volume Of Cuboid and Cube

Let’s discuss the volume of solids. If an object is solid, then the space occupied by such an object is measured and is termed the Volume of the object. On the other hand, if the object is hollow, then the interior is empty and can be filled with air, or some liquid that will take the shape of its container. In this case, the volume of the substance that can fill the interior is called the capacity of the container.

In short, the volume of an object is the measure of the space it occupies, and the capacity of an object is the volume of substance its interior can accommodate. Hence, the unit of measurement of either of the two is cubic unit.

First, we will consider the volume of the cuboid. Let us consider some rectangles stacked on top of each other. Suppose we say that the area of each rectangle is A, the height up to which the rectangles are stacked is h and the volume of the cuboid is V. Can you tell what would be the relationship between V, A, and h?

Volume of cuboid

The area of the plane region occupied by each rectangle × height = Measure of the space occupied by the cuboid

So, we get A × h = V

Volume of cuboid

That is, Volume of a Cuboid = base area × height = length × breadth × height
or l × b × h, where l, b and h are respectively the length, breadth and height of the cuboid.

Volume of cuboid

Similarly in cube volume is area into height. And we also know that all three dimensions of cube are equal so we have Volume of a Cube = edge × edge × edge = a cube where a is the edge of the cube.

Solved Examples

1. A wall of length 10 m was to be built across open ground. The height of the wall is 4 m and the thickness of the wall is 24 cm. If this wall is to be built up with bricks whose dimensions are 24 cm × 12 cm × 8 cm, how many bricks would be required?

Solution: Since the wall with all its bricks makes up the space occupied by it, we need to find the volume of the wall, which is nothing but a cuboid. Here, Length = 10 m = 1000 cm
Thickness = 24 cm
Height = 4 m = 400 cm
Therefore, Volume of the wall = length × thickness × height
= 1000 × 24 × 400 cm3

Now, each brick is a cuboid with length = 24 cm, breadth = 12 cm and height = 8 cm
So, volume of each brick = length × breadth × height
= 24 × 12 × 8 cm3

Volume of cuboid

2. A child playing with building blocks, which are of the shape of cubes, has built a structure. If the edge of each cube is 3 cm, find the volume of the structure built by the child.

Solution: Volume of each cube = edge × edge × edge = 3 × 3 × 3 cm cube = 27 cm3
Number of cubes in the structure = 15
Therefore, volume of the structure = 27 × 15 cm3
= 405 cm3

Open chat