Class 10 NCERT Maths Chapter 12: Surface Areas and Volumes
This chapter extends the concept of mensuration to three-dimensional objects. We learn to calculate the surface areas and volumes of different solids, and also how to solve problems involving conversion from one solid to another.
1. Surface Area Formulas
| Solid Shape | Surface Area Formula | Notes |
|---|---|---|
| Cube (side = \(a\)) | \(6a^2\) | All faces are squares. |
| Cuboid (\(l \times b \times h\)) | \(2(lb + bh + hl)\) | Opposite faces are equal. |
| Cylinder (radius \(r\), height \(h\)) | Curved: \(2\pi r h\) Total: \(2\pi r(h + r)\) |
Two circular bases and curved surface. |
| Right Circular Cone (radius \(r\), slant height \(l\)) | Curved: \(\pi r l\) Total: \(\pi r (l + r)\) |
\(l = \sqrt{h^2 + r^2}\) |
| Sphere (radius \(r\)) | \(4\pi r^2\) | Perfectly symmetrical. |
| Hemisphere (radius \(r\)) | Curved: \(2\pi r^2\) Total: \(3\pi r^2\) |
Has a flat circular base. |
2. Volume Formulas
| Solid Shape | Volume Formula |
|---|---|
| Cube (side = \(a\)) | \(a^3\) |
| Cuboid (\(l \times b \times h\)) | \(l \times b \times h\) |
| Cylinder | \(\pi r^2 h\) |
| Right Circular Cone | \(\frac{1}{3} \pi r^2 h\) |
| Sphere | \(\frac{4}{3} \pi r^3\) |
| Hemisphere | \(\frac{2}{3} \pi r^3\) |
3. Frustum of a Cone
A frustum is formed when a cone is cut by a plane parallel to its base and the upper part is removed.
- Slant height: \(l = \sqrt{h^2 + (R - r)^2}\)
- Curved surface area: \(\pi (R + r)l\)
- Total surface area: \(\pi (R + r)l + \pi R^2 + \pi r^2\)
- Volume: \(\frac{1}{3} \pi h (R^2 + Rr + r^2)\)
Here \(R\) = larger radius, \(r\) = smaller radius, \(h\) = height.
4. Conversion of Solids
When a solid is melted and recast into another solid, the volume remains the same.
Example: \[ \text{Volume of original solid} = \text{Volume of new solid(s)} \]
Key Points to Remember
- Use the same units for all dimensions before applying formulas.
- \(\pi\) is taken as \(\frac{22}{7}\) or \(3.14\) unless otherwise mentioned.
- Know the difference between curved surface area (CSA) and total surface area (TSA).
- Draw and label diagrams to avoid mistakes.