Formula for Surface Area of Cube – Derivation & Examples

Here you will learn formula for surface area of cube (total surface area and lateral surface area), its derivation and examples.

Let’s begin –

What is Cube ?

A cube is a cuboid which has all its edges or sides equal i.e length = breadth = height = s (side of cube).

Formula for Surface Area of Cube

(a) Total Surface Area of Cube (T.S.A)

T.S.A = \(6s^2\)

where s is the side of cube.

(b) Lateral Surface Area of Cube (L.S.A)

L.S.A = \(4s^2\)

where s is the side of cube.

Also Read : Formula for Volume of Cube – Derivation & Examples

Derivation :

Since a cuboid, whose length, breadth and the height are all equal, is called a cube. If each edge or side of the cube is s, then the total surface area of this cube would be 2(s × s + s × s + s × s), i.e., \(6s^2\).

Suppose, out of the six faces of a cube, we only find the area of the four faces, leaving the bottom and top faces. In such a case, the area of these four faces is called lateral surface area of the cube. So, lateral surface area of a cube of side s is equal to \(4s^2\). 

Example : Hameed has built a cubical water tank with lid for his house, with each outer edge 1.5 m long. He gets the outer surface of tank excluding the base, covered with square tiles of side 25 cm. Find how much he would spend for 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 or side of the cubical tank = 1.5 m = 150 cm (= s)

So, the surface area of tank = 5 × 150 × 150 \(cm^2\)

Area of each square of tile = side × side = 25 × 25 \(cm^2\)

So, the number of tiles required = surface area of the tank/area of each tile

=\(5 × 150 × 150\over 25 × 25\) = 180

Cost of 1 dozen tiles, i.e., cost of 12 tiles = 360

Therefore, the cost of one tile = 360/12 = 30

So, the cost of 180 tiles = 180 × 30 = Rs 5400

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