Chapter 12: Surface Areas and Volumes
Surface Area of Combined Solids
This section focuses on calculating the total surface area of objects made by joining basic 3D shapes such as cylinders, cones, spheres, and hemispheres. Since combined solids lose some internal surface area at the joints, the surface area is computed by summing only the curved or exposed areas. Several real-life examples are given, like toy tops and decorative blocks, to show how to combine and subtract areas from the shapes involved (e.g., cone + hemisphere, cube + hemisphere, cylinder + cone).
Volume of Combined Solids
In contrast to surface area, the volume of a composite solid is simply the sum of volumes of its parts, as the entire mass is retained. Examples include calculating air volume in a shed (cuboid + half-cylinder) or a toy made of a cone on top of a hemisphere. These problems use standard volume formulas and often include subtractions (e.g., reducing the capacity of a glass by a raised hemispherical base) to find actual usable space.
Applications and Problem Solving
The chapter includes diverse real-world problems involving solids like capsules, tents, poles, pen stands, and vessels with cavities or added parts. Students apply formulas for surface area and volume to practical designs and analyze how volume changes with added or removed sections. Exercises reinforce concepts by combining solids in various orientations, applying unit conversions, and interpreting results in everyday contexts like painting, packaging, and material use.