📚 Class 8 CBSE – Mathematics – Mensuration
(Prepared by an expert teacher – easy to read, clear and concise)
1. Simple Explanation
Mensuration is the branch of mathematics that deals with measuring the size of geometric figures:
- Linear measurement – length, breadth, height, radius, diameter.
- Area – flat surfaces.
- Perimeter (or Circumference) – the total length of the boundary.
- Volume – the amount of space a 3‑D figure occupies.
- Surface Area – the total area of all the outer faces of a 3‑D figure.
All these are obtained from basic formulas that relate the dimensions of the figure to the quantity we want to find.
2. Key Points & Formulas
| Figure | Dimensions | Perimeter / Circumference | Area (2‑D) | Volume | Surface Area |
|---|---|---|---|---|---|
| Rectangle | length (l), breadth (b) | (P = 2(l+b)) | (A = l \times b) | – | – |
| Right Triangle | sides (a,b,c) (c hypotenuse) | – | (A = \frac{1}{2}ab) | – | – |
| Circle | radius (r) | (C = 2\pi r) | (A = \pi r^2) | – | – |
| Cube | edge (a) | – | (A = 6a^2) | (V = a^3) | (SA = 6a^2) |
| Cuboid | length (l), breadth (b), height (h) | – | – | (V = lbh) | (SA = 2(lb + bh + hl)) |
| Sphere | radius (r) | – | – | (V = \frac{4}{3}\pi r^3) | (SA = 4\pi r^2) |
| Cylinder | radius (r), height (h) | – | – | (V = \pi r^2 h) | (SA = 2\pi r(h + r)) |
| Cone | radius (r), height (h), slant height (l) | – | – | (V = \frac{1}{3}\pi r^2 h) | (SA = \pi r(r + l)) |
| Right Circular Cone | radius (r), height (h), slant height (l=\sqrt{r^2+h^2}) | – | – | (V = \frac{1}{3}\pi r^2 h) | (SA = \pi r(r + l)) |
| Right Circular Cylinder | radius (r), height (h) | – | – | (V = \pi r^2 h) | (SA = 2\pi r(h+r)) |
| Right Circular Pyramid | base area (B), height (H) | – | – | (V = \frac{1}{3}BH) | – |
| Regular Polygon (n sides) | side (s), apothem (a) | (P = ns) | (A = \frac{1}{2} \times \text{Perimeter} \times a = \frac{1}{2}na s) | – | – |
Notes
- (\pi) is approximately (3.14).
- For a cone, the slant height (l) is the hypotenuse of the right triangle formed by radius, height, and slant height: (l = \sqrt{r^2 + h^2}).
- For a sphere, the diameter (d = 2r).
- In many problems, you may need to first find a missing dimension (e.g., find (l) for a cone) before plugging it into the formula.
3. Worked Example
Problem
A right circular cylinder has a radius of (5\text{ cm}) and a height of (12\text{ cm}).
- Find its volume.
- Find its total surface area.
Step‑by‑Step Solution
1. Volume
[ V = \pi r^2 h ]
Insert (r = 5\text{ cm}) and (h = 12\text{ cm}):
[ \begin{aligned} V &= 3.14 \times (5)^2 \times 12 \ &= 3.14 \times 25 \times 12 \ &= 3.14 \times 300 \ &= 942\ \text{cm}^3 \text{ (approx.)} \end{aligned} ]
2. Surface Area
[ SA = 2\pi r (h + r) ]
Insert the values:
[ \begin{aligned} SA &= 2 \times 3.14 \times 5 \times (12 + 5) \ &= 6.28 \times 5 \times 17 \ &= 31.4 \times 17 \ &= 533.8\ \text{cm}^2 \text{ (approx.)} \end{aligned} ]
Answer
- Volume ≈ 942 cm³
- Surface Area ≈ 534 cm² (rounded to nearest whole number)
4. Quick Revision Summary
| Concept | Formula | What it Gives |
|---|---|---|
| Perimeter of a rectangle | (P = 2(l+b)) | Boundary length |
| Area of a rectangle | (A = lb) | Flat area |
| Circumference of a circle | ( |