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Surface Area of a Square Prism

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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Definition

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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.

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 :

Surface Area of a Square Prism

Above figure shows a rectangular box with back and front faces as squares. Thus, this is called a square prism.

Formula

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The surface area of a square prism is the area of all its outer surfaces. The term "surface area" usually refers to the total surface area including bases. On the other hand, the surface area of side surfaces or walls is known as lateral surface area.

Lateral Surface Area

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
LSA of square prism = 4 a h,

where a is the side of the square base and h is the height of the prism.
Total Surface Area

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 :

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

How to Calculate Surface Area of a Square Prism

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

Step 1 : Determine the height "h" of the prism. It would be given in the question or in the diagram. Its unit is the unit of length, i.e. cm, m, inch, mm etc.

Step 2 : Find the perimeter of the base by multiplying the side of the square base by 4. Unit of perimeter will be the unit of length.

Step 3 : Calculate the product of perimeter of base and height of the prism which gives lateral surface area. The unit will be unit$^{2}$

Step 4 : If further, the total surface area is asked, we need to find the area of the base. This is done by squaring the side of the square base.

Step 4 : Take double to an area of the base and add it to the lateral surface area in order to get the total surface area. Represent it with appropriate unit which is unit$^{2}$.

Examples

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Have a look at the following examples.
Example 1 : Calculate the surface area of the square prism of height 10 inches and each side of the square base being 4.5 inches.

Solution : Height of prism, h = 10 in

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$^{2}$
Example 2 : Calculate the lateral surface area of a square prism whose side of the base is 14.5 cm and height is 22.5 cm.

Solution : 
Height of prism, h = 14.5 cm

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$^{2}$
Example 3 : Determine the side of a square prism whose all edges are equal and its surface area is 864 sq cm.

Solution : Since all edges are equal, therefore we have

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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