Class 10 Maths Notes Chapter 12 — Surface Areas and Volumes#
Formulas for basic solids#
| Solid | Curved / lateral surface area | Total surface area | Volume |
|---|---|---|---|
| Cuboid ($l \times b \times h$) | $2h(l + b)$ | $2(lb + bh + hl)$ | $lbh$ |
| Cube (edge $a$) | $4a^2$ | $6a^2$ | $a^3$ |
| Cylinder ($r$, $h$) | $2\pi rh$ | $2\pi r(r + h)$ | $\pi r^2 h$ |
| Cone ($r$, $h$, $l$) | $\pi rl$ | $\pi r(r + l)$ | $\frac{1}{3}\pi r^2 h$ |
| Sphere ($r$) | $4\pi r^2$ | $4\pi r^2$ | $\frac{4}{3}\pi r^3$ |
| Hemisphere ($r$) | $2\pi r^2$ | $3\pi r^2$ | $\frac{2}{3}\pi r^3$ |
Slant height of a cone:
$$l = \sqrt{r^2 + h^2}$$Surface area of a combination#
Add the areas of only the surfaces that are visible (exposed). Faces that are joined together, or covered, are not counted.
| Solid | Surface area |
|---|---|
| Cone on a hemisphere (toy) | CSA of cone + CSA of hemisphere |
| Cylinder with a hemisphere at each end (capsule) | CSA of cylinder + 2 × CSA of hemisphere |
| Cylinder with a cone on top (tent, no base) | CSA of cylinder + CSA of cone |
| Hemisphere on a cube | TSA of cube − base of hemisphere + CSA of hemisphere |
| Hemisphere scooped out of a cube | TSA of cube − base of hemisphere + CSA of hemisphere |
| Cone hollowed out of a cylinder | CSA of cylinder + one base + CSA of cone |
- Note: area of the joined face is not subtracted twice. For a hemisphere on a cube, you remove the circle from the cube’s top face, then add the curved hemisphere.
Volume of a combination#
Volume of a combined solid = sum of the volumes of its parts. If one solid is cut out of another, subtract its volume.
- Volume is never “partly hidden”, so unlike surface area, every part counts.
- Height of the middle part: subtract the heights of the end parts from the total. A capsule 14 mm long with radius 2.5 mm has a cylinder of length $14 - 2 \times 2.5 = 9$ mm.
- Displacement: when objects are dropped into a full container, the volume of water that overflows equals the volume of the objects.
- Mass = volume × density (mass of 1 cm³).
Common mistakes#
- Adding the total surface areas of both solids, so joined faces get counted.
- Using the full height of the combined solid as the height of one part.
- Using the slant height $l$ in a volume formula. Volume uses the vertical height $h$.
- Using the diameter as the radius.
- Mixing units (mm, cm, m; cm³ and litres: $1000 \text{ cm}^3 = 1$ litre).