Chapter 12 — Surface Areas and Volumes

Chapter 12 of the Class 10 Maths NCERT textbook, "Surface Areas and Volumes", covers solids formed by combining basic shapes such as cuboids, cones, cylinders, spheres, and hemispheres, teaching students to find the combined surface area from the visible parts and the combined volume by summing the constituent solids.

• Combining basic solids — Real objects are often built by joining simple solids — a cone on a hemisphere makes a toy, a cylinder with two hemispheres makes an oil tanker. The chapter treats these composite shapes as sums of familiar parts.
• Surface area counts only what shows — When solids are joined, the total surface area adds up only the curved surfaces of the exposed, visible parts — the hidden faces where pieces meet do not count.
• Volume simply adds — Unlike surface area, the volume of a combined solid is always the straightforward sum of its individual volumes. Even when a shape is hollowed out, the exposed inner surface adds to the total surface area.

1. 01The total surface area of a combined solid equals the sum of the curved surface areas of only the exposed (visible) parts of each constituent solid.
2. 02The volume of a combined solid is always the direct sum of the volumes of its individual constituent solids.
3. 03Common composite shapes include: cylinder with two hemispheres (oil tankers, capsules), cone surmounted by hemisphere (toys, tops), cuboid surmounted by hemisphere (decorative blocks), and cuboid surmounted by half cylinder (industrial sheds).
4. 04For the playing top (cone surmounted by hemisphere, total height 5 cm, diameter 3.5 cm), the surface area to be coloured is approximately 39.6 cm².
5. 05For a decorative cube-and-hemisphere block (cube edge 5 cm, hemisphere diameter 4.2 cm), the total surface area is 163.86 cm².
6. 06When a solid is hollowed out (e.g., hemisphere scooped from cylinder ends), the exposed inner curved surface adds to — not subtracts from — the total surface area.