Mensuration rewards memory and care: know the formula, use the radius (not the diameter), and keep your units straight.
| Shape | Area / Volume |
|---|---|
| Rectangle | length × breadth |
| Square | side × side |
| Triangle | 1/2 × base × height |
| Circle | pi × r^2 (circumference 2 × pi × r) |
| Cube | volume = side^3 |
| Cuboid | volume = l × b × h |
| Cylinder | volume = pi × r^2 × h |
How to Approach It
- Identify the shape and the quantity — decide whether you need area, perimeter, surface area or volume, and recall the matching formula before plugging in numbers.
- Feed the correct measurement — use the radius, not the diameter, in circle formulas, and choose pi = 22/7 when the radius is a multiple of 7 so the arithmetic stays whole.
- Apply the scaling rule — if every length is multiplied by k, area scales by k^2 and volume by k^3. This answers 'by what factor' questions with no numbers at all.
- Decompose odd figures — break a composite shape into rectangles, triangles and circles, then add or subtract their areas. A quick sketch keeps the pieces straight.
Techniques & Methods
- Right formula, right input — always feed the radius (not the diameter) into circle formulas.
- Scaling by k^2 and k^3 — multiply every length by k → area scales k^2, volume k^3. e.g. double a side → area becomes 4×.
- Pick pi = 22/7 when radius is a multiple of 7 — the 7s cancel and the arithmetic stays whole. e.g. r = 7 → area = 154.
- Split composite shapes — add or subtract simple shapes to handle an odd figure.
The Edge
Scaling shortcut: if every linear dimension is multiplied by k, area scales by k^2 and volume by k^3. Double a square's side and its area becomes 4×; double a cube's edge and its volume becomes 8×. This clears every "by what factor" question without plugging in numbers.Worked example
Find the area of a circle of radius 7 (use pi = 22/7).
- Identify the shape and the quantity asked: the area of a circle, which is pi × r².
- Choose pi = 22/7 here because the radius 7 will cancel the 7 in the denominator, keeping the arithmetic whole.
- Substitute: 22/7 × 7² = 22/7 × 49.
- The 7 cancels into the 49 (leaving 7): 22 × 7 = 154 square units.
Answer: Area = 154 sq units
Worked example
Find the volume of a cube of side 4.
- A cube has all edges equal, and its volume is side × side × side, i.e. side³.
- Here the side is 4, so volume = 4³.
- Compute: 4 × 4 × 4 = 64.
- So the volume is 64 cubic units. (Volume is in cubic units, unlike area, which is in square units.)
Answer: Volume = 64 cubic units
Worked Drills
Worked example
The area of a triangle with sides 13, 14 and 15 is:
- Use Heron's formula with s = (13+14+15)/2 = 21.
- Area = sqrt(21 × (21−13) × (21−14) × (21−15)) = sqrt(21 × 8 × 7 × 6).
- That equals sqrt(7056) = 84.
Answer: b) 84
Worked example
The diagonal of a square is 10 × sqrt(2). Its area is:
- Diagonal = side × sqrt(2), so side = 10.
- Area = side² = 10² = 100.
Answer: b) 100
Worked example
The curved surface area of a cylinder with r=7, h=10 (pi=22/7) is:
- CSA = 2 × pi × r × h = 2 × 22/7 × 7 × 10.
- The 7 cancels: 2 × 22 × 10 = 440.
Answer: a) 440
Worked example
A wire bent into a square of side 11 is rebent into a circle. The radius (pi=22/7) is:
- Wire length = perimeter of square = 4 × 11 = 44.
- This becomes the circumference: 44 = 2 × 22/7 × r.
- Solving gives r = 7.
Answer: a) 7
Worked example
The volume of a sphere of radius 3 (pi=22/7) is approximately:
- Volume = (4/3) × pi × r³ = (4/3) × 22/7 × 27.
- Compute: ≈ 113.14.
Answer: a) 113.14
Worked example
The ratio of the areas of two circles with radii 3 and 4 is:
- Area scales as the square of the radius.
- Ratio = 3² : 4² = 9 : 16.
Answer: b) 9:16
Worked example
Circumference of a circle with r = 7 (pi = 22/7):
- Circumference = 2 × pi × r = 2 × 22/7 × 7.
- The 7 cancels: 2 × 22 = 44.
Answer: b) 44
Worked example
Volume of a cylinder r = 7, h = 10 (pi = 22/7):
- Volume = pi × r² × h = 22/7 × 49 × 10.
- The 7 cancels into 49 (leaving 7): 22 × 7 × 10 = 1540.
Answer: b) 1540
⚠ Watch out
- Use the radius, not the diameter, in circle formulas.
- Area is in square units, volume in cubic units — don't mix them.
- For combined shapes, add or subtract areas carefully; sketch it first.
Takeaways
- Memorise the core shape formulas and decide area vs volume before computing.
- Always plug in the radius, and pick pi = 22/7 when the radius is a multiple of 7.
- Scaling: multiply lengths by k → area ×k^2, volume ×k^3, answering factor questions instantly.
- Break composite figures into simple shapes and add or subtract their areas.