12.3. Volume of a Combination of Solids

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.

This chunk introduces the concept of calculating the volumes of combined solids. While previously we learned how to find the surface areas of solids made from basic shapes (like spheres or cylinders), here we focus on how to combine their volumes. It emphasizes that when calculating volume, we don’t lose any part of the volume as we did with surface areas—when two solids join, we simply add their volumes together.

Think about filling up a jar with different types of marbles. If you have a red marble (say a sphere) and a green marble (a cube), the total amount of space taken up by both marbles in the jar is simply the sum of the spaces they occupy individually.

Example 5: Shanta runs an industry in a shed which is in the shape of a cuboid surmounted by a half cylinder (see Fig. 12.12). If the base of the shed is of dimension 7 m × 15 m, and the height of the cuboidal portion is 8 m, find the volume of air that the shed can hold...

This example outlines a practical scenario where you calculate the volume of air in a shed shaped like a cuboid with a half cylinder on top. The volume is calculated by adding the volume of the cuboidal part (length × breadth × height) to the volume of the half cylinder (which is half of the volume of a full cylinder). This illustrates the direct application of volume calculations for combined shapes, demonstrating that their total volume is simply the sum of each part’s volume.

Imagine the shed as a large food storage container. If you know the area of the floor and the height of the walls, you can easily calculate how much food it can store. By adding the extra storage area provided by the half-cylinder top, you're maximizing the amount of food that can be safely kept in this container.

Example 6: A juice seller was serving his customers using glasses as shown in Fig. 12.13. The inner diameter of the cylindrical glass was 5 cm, but the bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass...

In this example, we analyze a glass with a unique shape: a cylinder with a hemispherical base. The apparent capacity (the full volume the glass would hold if it were a simple cylinder) is calculated first, followed by subtracting the volume of the hemispherical portion since it takes up space inside the glass. This process illustrates the necessity of accounting for internal shapes while calculating volume.

Think of the glass as a piggy bank that has a small dome at the bottom. If you want to know how much money you can fit inside, you must first figure out how much space the whole piggy bank has, and then subtract the space taken up by the dome at the bottom to determine how much cash you can actually deposit.

Example 7: A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm...

This example walks through the process of calculating the volume of a toy shaped like a cone on top of a hemisphere. The volume of both shapes is calculated separately, utilizing the formulas for the volume of a cone and a hemisphere, and added together. This displays how different shapes can interact to define a larger object’s volume.

Visualize a birthday hat that combines a pointed cone shape with a round base that looks like half of a ball. If you wanted to know how much space that hat or toy occupies, you would measure the total space taken by both the cone and the hemisphere, which is just as straightforward as stacking boxes on top of each other to determine their combined height.

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