12. Surface Areas and Volumes

From Class IX, you are familiar with some of the solids like cuboid, cone, cylinder, and sphere. You have also learnt how to find their surface areas and volumes. In our day-to-day life, we come across a number of solids made up of combinations of two or more of the basic solids as shown above. You must have seen a truck with a container fitted on its back, carrying oil or water from one place to another. Is it in the shape of any of the four basic solids mentioned above? You may guess that it is made of a cylinder with two hemispheres as its ends.

In this section, we introduce some common three-dimensional shapes that we deal with in mathematics: cuboids, cones, cylinders, and spheres. We revisit how to calculate their surface areas and volumes. We also observe that in real life, we often encounter combined shapes, such as cylindrical tanks with hemispherical ends. Understanding how to break down these shapes into familiar solids will be important for calculating their surface areas and volumes.

Think of a truck carrying water. The container is not just a simple shape but is instead likely a combination of a cylinder and two hemispheres. This is similar to how we build complex structures in reality, where various shapes come together.

Let us consider the container seen in Fig. 12.2. How do we find the surface area of such a solid? 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. The total surface area of the new solid is the sum of the curved surface areas of each of the individual parts.

To calculate the surface area of a combined shape, we need to identify its individual components. For instance, if our solid consists of a cylinder with two hemispherical ends, we calculate the curved surface area of the cylinder (CSA of cylinder) and the curved surface areas of both hemispheres. The formula we use is as follows: Total Surface Area = CSA of Hemisphere 1 + CSA of Cylinder + CSA of Hemisphere 2. This method allows us to systematically find the surface area by referencing simpler problems we have already solved.

Imagine you are wrapping a gift that has an unusual shape, like a packed lunch container that is a cylinder with round ends. Before wrapping, you break down the surfaces: how much paper is needed to cover the curved part and how much for each rounded end. This is just like calculating the surface area for complex shapes!

Example 1: Rasheed got a playing top (lattu) shaped like a cone surmounted by a hemisphere. The entire top is 5 cm in height and the diameter of the top is 3.5 cm. The total surface area of the toy is equal to the CSA of the hemisphere plus the CSA of the cone.

In this example, we see that the top is made up of two solid shapes: a cone on top of a hemisphere. To find the total surface area that Rasheed needs to paint, we calculate the curved surface area of the hemisphere and the cone. First, remember that the height of the cone is not just 5 cm; we need to subtract the radius of the hemisphere to find the height of the cone alone. Once we calculate each surface area using known formulas, we can add them to get the total.

Think of customizing your favorite dessert. If you have a cupcake (the cone) topped with frosting shaped like a half-ball (the hemisphere), you would want to know how much frosting you need to cover the entire surface. That’s just like figuring out how much paint Rasheed requires for his toy!

Example 2: The decorative block shown is made of two solids — a cube and a hemisphere. The base of the block is a cube with edge 5 cm, and the hemisphere on the top has a diameter of 4.2 cm.

In this example, we start by calculating the surface area of the cube: the formula for the total surface area of a cube is 6 times the area of one face (edge square). However, we must remember that the area of the base where the hemisphere is attached does not count towards the surface area. Instead, we subtract this area and then add the curved surface area of the hemisphere. This calculation requires careful attention to the shapes involved and how they connect.

Imagine a cake that is a cube with a dome on top, similar to a cake with a hemispherical decoration on it. When you frost this dessert, you wouldn’t frost the part where the dome meets the cake. Thus, knowing how to adjust the surface area calculations mirrors our real-life experience with customized cakes!

In the previous section, we have discussed how to find the surface area of solids made up of a combination of two basic solids. Here, we shall see how to calculate their volumes. In calculating the volume, we sum the volumes of the individual components without subtracting any overlapping parts.

Unlike surface area calculations, when finding volumes for combined shapes (like a cylinder and a cone), we simply add the volumes of each shape. Here, we use the formulas for volume (Volume of cylinder = πr²h; Volume of cone = 1/3πr²h) to find the total volume correctly. This is because the shapes keep the full volume when combined; nothing is hidden like with surface areas.

Think of a glass filled with ice cubes and liquid. The total volume inside the glass is the sum of the liquid volume and the volumes of the ice cubes. Whether the ice is present or not, the glass still holds the same overall volume of space. This analogy helps us understand that when calculating volumes, we measure everything combined!

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