12.2. Surface Area of a Combination of Solids

Let us consider the container seen in Fig. 12.2. How do we find the surface area of such a solid? Now, whenever we come across a new problem, we first try to see, if we can break it down into smaller problems, we have earlier solved. We can see that this solid is made up of a cylinder with two hemispheres stuck at either end. It would look like what we have in Fig. 12.4, after we put the pieces all together.

To find the surface area of a complex solid, we can dissect it into simpler shapes that we already know how to calculate the surface area for. In this example, we have a container that resembles a cylinder with two hemispherical ends. Recognizing the individual shapes allows us to apply our existing knowledge about calculating the surface area of a cylinder and a hemisphere.

Think of it like building a model of a house using blocks. You wouldn't try to paint the whole house at once; instead, you can paint each block one at a time, applying the same principles of color to each section.

If we consider the surface of the newly formed object, we would be able to see only the curved surfaces of the two hemispheres and the curved surface of the cylinder. So, the total surface area of the new solid is the sum of the curved surface areas of each of the individual parts. This gives,

TSA of new solid = CSA of one hemisphere + CSA of cylinder + CSA of other hemisphere where TSA, CSA stand for ‘Total Surface Area’ and ‘Curved Surface Area’ respectively.

The total surface area (TSA) of the new solid is calculated by summing the curved surface areas (CSA) of each distinct solid that makes it up. For our solid with cylindrical and hemispherical sections, we include only the curved surface areas because the flat surfaces where they join are not visible.

Imagine you are wrapping a birthday gift with shiny wrapping paper. You would only cover the visible sides of the box and the bow on top, ignoring the parts that are already covered or stuck together.

Let us now consider another situation. Suppose we are making a toy by putting together a hemisphere and a cone. Let us see the steps that we would be going through.

First, we would take a cone and a hemisphere and bring their flat faces together. Here, of course, we would take the base radius of the cone equal to the radius of the hemisphere, for the toy is to have a smooth surface. So, the steps would be as shown in Fig. 12.5.

When combining a hemisphere and a cone, we align their flat faces to ensure they fit together smoothly. It is essential that the radius of the cone’s base matches that of the hemisphere so there are no gaps. This combination creates a solid that collectively has a new surface area calculated from both components.

Think of making a smoothie. Just like blending different fruits together creates a new flavor, combining a cone and a hemisphere into one solid shape creates a new toy design!

Now if we want to find how much paint we would require to colour the surface of this toy, what would we need to know? We would need to know the surface area of the toy, which consists of the CSA of the hemisphere and the CSA of the cone.

So, we can say: Total surface area of the toy = CSA of hemisphere + CSA of cone.

To determine how much paint is needed for the toy, we first need to calculate its surface area. This requires knowing the curved surface area of the hemisphere and the cone. By summing these two areas, we find the total area on which paint will be applied.

Imagine getting a canvas ready for painting. You first measure the area to know how much paint you need. The same concept applies here: we measure the surface area of our toy to know how much paint to buy.

Example 1: Rasheed got a playing top (lattu) as his birthday present, which surprisingly had no colour on it. He wanted to colour it with his crayons. The top is shaped like a cone surmounted by a hemisphere (see Fig 12.6). The entire top is 5 cm in height and the diameter of the top is 3.5 cm. Find the area he has to colour. (Take π = 22/7)

This example illustrates the method of applying the concepts discussed. Here, Rasheed's top combines a cone and a hemisphere, and we can determine the surface area he needs to color by using the known dimensions. We calculate the curve surfaces separately and sum them up, leveraging our understanding of the shapes involved.

Think of Rasheed's top like making your own ice cream sundae. You have the cone for the ice cream and the flat bowl on top. When determining the cost of toppings, you first need to measure both components to make sure you have enough!

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