Concept 1
First identify the dimension being measured
Perimeter is boundary length. Area covers a plane region. Surface area covers the exposed faces or curved skin of a solid. Volume measures occupied space.
| Measure | Units | Scale factor k |
|---|---|---|
| Length / perimeter | cm | k |
| Area / surface area | cm² | k² |
| Volume | cm³ | k³ |
Conversion: 1 m=100 cm, but 1 m²=10,000 cm² and 1 m³=1,000,000 cm³.
Concept 2
Plane figures use a perpendicular height
| Figure | Perimeter | Area |
|---|---|---|
| Rectangle l,b | 2(l+b) | lb |
| Square a | 4a | a²; diagonal=a√2 |
| Parallelogram | 2(a+b) | base×height |
| Rhombus | 4a | ½d₁d₂ |
| Trapezium | sum of sides | ½(sum of parallel sides)×height |
A sloping side is not the height unless it is perpendicular to the chosen base.
Concept 3
Choose the triangle formula from the data
- Base b and height h: A=½bh.
- Sides a,b,c: s=(a+b+c)/2 and A=√[s(s−a)(s−b)(s−c)].
- Equilateral side a: A=(√3/4)a²; height=(√3/2)a.
- Right-triangle legs p,q: A=½pq.
Concept 4
Radius controls circular length and area
For radius r, circumference=2πr and area=πr². A semicircle's perimeter is πr+2r because its diameter is part of the boundary.
| Part | Measure |
|---|---|
| Arc θ° | (θ/360)×2πr |
| Sector θ° | (θ/360)×πr² |
| Ring, radii R,r | π(R²−r²) |
Use the value of π stated in the question. Otherwise retain π until the end or use 22/7 when the numbers suit it.
Concept 5
Composite area is addition or subtraction
For a width x path outside an l×b rectangle, outer dimensions are (l+2x) and (b+2x). Path area=(l+2x)(b+2x)−lb. For an inner path, the remaining dimensions are (l−2x) and (b−2x).
Split an irregular region into non-overlapping known shapes, or calculate an enclosing area and subtract its holes.
Concept 6
A prism keeps the same cross-section
| Solid | Lateral / curved area | Total area | Volume |
|---|---|---|---|
| Cuboid l,b,h | 2h(l+b) | 2(lb+bh+hl) | lbh |
| Cube a | 4a² | 6a² | a³ |
| Cylinder r,h | 2πrh | 2πr(h+r) | πr²h |
Any prism has volume=cross-sectional area×perpendicular length. For open tanks or pipes, include only surfaces that actually exist.
Concept 7
Separate a cone's vertical and slant heights
For a right circular cone, ℓ=√(r²+h²), curved area=πrℓ, total area=πr(ℓ+r), and volume=⅓πr²h.
For a frustum with radii R,r and vertical height h, ℓ=√[h²+(R−r)²], curved area=π(R+r)ℓ, and volume=⅓πh(R²+Rr+r²).
Concept 8
Spheres depend only on radius
| Solid | Area | Volume |
|---|---|---|
| Sphere | 4πr² | ⁴⁄₃πr³ |
| Hemisphere | 2πr² curved; 3πr² total | ⅔πr³ |
The hemisphere's total area includes its circular base; curved area does not.
Concept 9
Capacity and recasting conserve volume
1 cm³=1 mL, 1000 cm³=1 L, and 1 m³=1000 L. When a solid is melted and recast without loss, old volume equals new volume. Liquid transferred between vessels also keeps its volume.
If every dimension is multiplied by k, area changes by k² and volume by k³.
Worked methods
Solve by naming the region first
Outer path
A 20 m×14 m lawn has a 2 m outer path. Outer size=24×18. Path area=432−280=152 m².
Heron's formula
Sides 13,14,15 give s=21. Area=√(21×8×7×6)=√7056=84 cm².
Sector
A 90° sector with r=14 cm has area=¼×(22/7)×14²=154 cm².
Cylinder capacity
For r=7 cm,h=10 cm, V=(22/7)×49×10=1540 cm³=1.54 L.
Cone
For r=3,h=4, ℓ=5. Curved area=15π cm² and volume=12π cm³.
Recasting
A sphere of radius 3 cm makes spheres of radius 1 cm. Number=(3/1)³=27.
Exam traps
Define the exposed measurement before calculating
Diameter used as radius
Convert d to r=d/2 first.
Linear unit conversion
Square and cube the conversion factor for area and volume.
Curved versus total
Add bases only when present and exposed.
Slant versus vertical
Cone area uses ℓ; volume uses h.
Path width counted once
An outer border changes each full dimension by 2x.
Early approximation
Keep π and roots exact until the final step.
Original practice
Check understanding after learning
Original syllabus-aligned questions—not claimed previous-year questions.
Rapid revision
The one-minute formula map
- Rectangle A=lb; triangle A=½bh; trapezium A=½(a+b)h.
- Heron: s=(a+b+c)/2; A=√[s(s−a)(s−b)(s−c)].
- Circle C=2πr; A=πr²; sector fraction=θ/360.
- Cuboid V=lbh; cube V=a³; cylinder V=πr²h.
- Cone ℓ²=r²+h²; CSA=πrℓ; V=⅓πr²h.
- Sphere SA=4πr²; V=⁴⁄₃πr³; hemisphere TSA=3πr².
- 1 cm³=1 mL; 1000 cm³=1 L; 1 m³=1000 L.
- Linear k gives area k² and volume k³; recasting conserves volume.
Source ledger
Official foundations
- SSC CGL official syllabus: Mensuration scope
- NCERT official textbook portal: areas, surface areas and volumes
- NCERT official exemplar directory
Explanations, worked examples and practice are original. Official references are credited and not reproduced. Drafted 29 September 2026.
Trigonometry and heights.