Chapter 23.2 - Surface area of Prisms

In the previous section we completed the discussion on volume of prisms. In this section we will see surface area of prisms.

Consider the triangular prism in fig.23.11(a) below. 
• Base of this prism is a triangle having sides 4, 5 and 6 cm. 
• Height of this prism is 8 cm

Fig.23.11

• But this prism is open at top and bottom. So it is like a tube.
• This tube is formed by three rectangles. They are:
    ♦ Rectangle with height 8 cm, width 4 cm
    ♦ Rectangle with height 8 cm, width 5 cm
    ♦ Rectangle with height 8 cm, width 6 cm

1. We know that there are 3 vertical edges between the above 3 rectangles
2. A cut is made along the edge between the first and third rectangle
3. Then the tube is opened. This is shown in fig.23.11(b)
4. Once we open it like that, we can spread it out on a flat surface. What we get is a perfect rectangle.
• The height of this rectangle is the same 8 cm of the prism
• The length of this rectangle is (4 +5 +6) = 15 cm
• So area of the rectangle = 15 × 8 = 120 cm²
5. But (4 +5 +6) is the perimeter of the base of the original prism 
6. So area of the rectangle = perimeter of the base of the original prism × height of the original prism
7. But from the fig.b, we can see that, area of this rectangle is the area of all the lateral surfaces of the original prism. So we get a formula for finding the lateral surface area of a triangular prism:
■ Lateral surface area of a triangular prism = Perimeter of the base triangle × height of the prism
[Note: In step (2) above, we made a cut along the edge between the first and third rectangle. In fact, we can make the cut along any one of the three vertical edges] 

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• Consider the case when the prism in fig.a is not a tube. 

• In that case, it will have two extra triangles. One at top and the other at bottom. These extra triangles are shown in blue colour in the fig.23.12 below. 

• Each of these triangles have sides 4, 5 and 6 cm. The ‘spreading out’ will be as shown in fig.23.12(b).

Fig.23.12

• In such a situation, we call it: ‘total surface area of the prism’.
• Obviously, it will be equal to: Lateral surface area + (2 × Area of base triangle)

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In the above discussion, we have considered triangular prism. We will now consider a quadrilateral prism.

• Consider the quadrilateral prism in fig.23.13(a) below. 
• Base of this prism is a quadrilateral having sides p, q, r and s cm. 
• Height of this prism is h cm

Fig.23.13

• But this prism is open at top and bottom. So it is like a tube.
• This tube is formed by 4 rectangles. They are:
    ♦ Rectangle with height h cm, width p cm
    ♦ Rectangle with height h cm, width q cm
    ♦ Rectangle with height h cm, width r cm
    ♦ Rectangle with height h cm, width s cm

1. We know that there are 4 vertical edges between the above 4 rectangles
2. A cut is made along the edge between the first and fourth rectangles
3. Then the tube is opened. This is shown in fig.23.13(b)
4. Once we open it like that, we can spread it out on a flat surface. What we get is a perfect rectangle.
• The height of this rectangle is the same h cm of the prism
• The length of this rectangle is (p +q +r +s)
• So area of the rectangle = (p +q +r +s)h
5. But (p +q +r +s) is the perimeter of the base of the original prism 
6. So area of the rectangle = perimeter of the base of the original prism × height of the original prism
7. But from the fig.b, we can see that, area of this rectangle is the area of all the lateral surfaces of the original prism. So we get a formula for finding the lateral surface area of a quadrilateral prism:
■ Lateral surface area of a quadrilateral prism = Perimeter of the base quadrilateral × height of the prism

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• Consider the case when the prism in fig.a is not a tube. 

• In that case, it will have two extra quadrilaterals. One at top and the other at bottom. These extra quadrilaterals are shown in blue colour in the fig.23.14 below.  

• Each of these quadrilaterals have sides p, q, r and s cm. The ‘spreading out’ will be as shown in fig.23.14(b).

Fig.23.14

• In such a situation, we call it: ‘total surface area of the prism’.
• Obviously, it will be equal to: Lateral surface area + (2 × Area of base quadrilateral)

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We have seen the cases of triangle and quadrilateral. We will get the same result for any polygonal prism. We can write it in the form of a theorem.

Theorem 23.2:

• We have a prism of height 'h' cm

• The base can be of any polygonal shape

• Area of the base is 'a' cm²

■ Then the lateral surface area = Perimeter of the base quadrilateral × height
■ Total surface area = Lateral surface area + 2a

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In the next section we will see a few solved examples.

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