Prisms are among the most common everyday shapes. Let us recall that a prism is a solid object having flat surfaces. Among them, two are similar and parallel which are known as bases and are used to name the kind of prism.

When you look around your house, classroom or neighborhood, you find several types of prisms, such as rectangular prisms,

square prisms, triangular prisms and they can be spotted very easily.

In today's chapter, we are going to throw light on the square prism which perhaps is the most outstanding geometrical shape and the most common type of prism. It makes out countless objects in our everyday life. We shall learn about formula and calculation of the surface area of a square prism in detail.

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A square prism is defined as a prism having two squares as bases. Remaining lateral surfaces can be parallelograms (oblique square prism) or rectangles or even squares (right square prism).

For example: Dice with all square surfaces, Rubik's cube, tissue box or cardboard boxes with two square faces.

For example:

A square prism does not necessarily have all surfaces to be squared. They should only have square bases. Therefore, it is to be noted that as long as a prism has two opposite faces as squares, it is categorized as a square prism which is demonstrated in the diagram below :

The formula for the lateral surface area of a prism is equal to the product to the perimeter of base and height of the prism.

LSA = Perimeter of base $\times$ Height

Here, the base is square shaped. So, perimeter of base is

P = 4 $\times$ side = 4 a

Hence

The formula for total surface area of a prism is :

TSA = LSA + 2 Area of base

Since area of square = (side)$^{2}$ = a$^{2}$

Hence, we may write total surface area of square prism is under :

While calculating the surface area of a square prism, the points mentioned below should be followed.

Have a look at the following examples.

Perimeter of base, P = 4 $\times$ side = 4 $\times$ 4.5 = 18 in

Area of base, A = side$^{2}$

= 4.5$^{2}$ = 20.25 in$^{2}$

Total surface area = P h + 2 A

TSA = 18 $\times$ 10 + 2 $\times$ 20.25

= 180 + 40.5 = 220.5

Surface area of the given prism is 220.5 in$^{2}$

Solution :

Perimeter of base, P = 4 $\times$ side

P = 4 $\times$ 14.5 = 58 cm

Lateral surface area = Perimeter of base $\times$ Height of prism

LSA = 58 $\times$ 22.5

= 1305

Therefore lateral surface area is 1305 cm$^{2}$

Height of the prism = edge = a (say)

i.e. h = a

Formula for total surface area is given as :

TSA = 4 a h + 2 a$^{2}$

864 = 4 a a + 2 a$^{2}$

864 = 4 a$^{2}$ + 2a$^{2}$

864 = 6 a$^{2}$

a$^{2}$ = $\frac{864}{6}$

a$^{2}$ = 144

a = 12

Side of the given prism is 12 cm.

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