Chapter 12 — Surface Areas and Volumes

NCERT Class 10 Maths Chapter 12 covers surface areas and volumes of solids formed by combining basic shapes such as cuboids, cones, cylinders, spheres, and hemispheres, teaching students to calculate both the combined surface area (using curved surface areas of visible parts) and the combined volume (by summing the volumes of constituent solids).

Chapter 12 of NCERT Class 10 Mathematics extends students' knowledge of surface areas and volumes from Class 9 to composite solids — objects formed by joining two or more basic solids such as cuboids, cones, cylinders, spheres, and hemispheres. The chapter explains that the total surface area of a combined solid is the sum of the curved surface areas of only the visible (exposed) parts, not the total surface areas of all constituents. In contrast, the volume of a combined solid is always the sum of the volumes of its individual parts. Real-life examples include oil tankers (cylinder with two hemispheres), toys (cone surmounted by hemisphere), bird-baths, and industrial sheds.

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.