Surface Area & Volume
Every shape has its own formula — learn each one as a concept block with an example.
A. 2D Shapes — Area & Perimeter
2D Shapes at a Glance
Square (side = s)
All four sides equal. Area = side × side. Perimeter = 4 × side.
Example — s = 6 cmArea = 6² = 36 cm² Perimeter = 4×6 = 24 cm
Rectangle (length × width)
Two pairs of equal sides. Area = l × w. Perimeter = 2(l + w).
Example — l=8, w=3Area = 8×3 = 24 cm² Perimeter = 2(8+3) = 22 cm
Triangle (base × height)
Area = ½ × base × height. Height must be perpendicular (not the slant side).
Example — base=10, height=6Area = ½ × 10 × 6 = 30 cm²
Circle (radius = r)
Area = πr². Circumference = 2πr. Use π = 22/7 when r is a multiple of 7.
Example — r = 7 cm, π = 22/7Area = 22/7 × 49 = 154 cm² Circumference = 2 × 22/7 × 7 = 44 cm
Trapezium (parallel sides a, b)
Two parallel sides of different lengths. Area = ½ × (a + b) × h.
Example — a=6, b=10, h=4Area = ½ × (6+10) × 4 = ½ × 64 = 32 cm²
B. 3D Shapes — Volume & Surface Area
Cube (side = s)
All six faces are identical squares. Volume = s³. TSA = 6s².
Example — s = 4 cmVolume = 4³ = 64 cm³ TSA = 6 × 16 = 96 cm²
Cuboid (l × w × h)
A rectangular box. Volume = l × w × h. TSA = 2(lw + wh + lh).
Example — l=5, w=3, h=2Volume = 30 cm³ TSA = 2(15+6+10) = 62 cm²
Cylinder (radius r, height h)
Two circular faces plus a curved surface. Volume = πr²h. TSA = 2πr(r + h).
Example — r=7, h=10, π=22/7Volume = 22/7 × 49 × 10 = 1540 cm³
Sphere (radius r)
Perfectly round solid. Volume = (4/3)πr³. TSA = 4πr².
Example — r=3, π=3.14Volume = (4/3) × 3.14 × 27 ≈ 113.1 cm³
Cone (radius r, height h, slant l)
Circular base tapering to a point. Volume = (1/3)πr²h. Slant height l = √(r² + h²).
Example — r=3, h=4 → l=5Volume = (1/3) × 3.14 × 9 × 4 ≈ 37.7 cm³
Example — Cylinder: r=7 cm, h=10 cm (π=22/7)
1Volume = πr²h = 22/7 × 7² × 10
2= 22/7 × 49 × 10 = 22 × 7 × 10 = 1540 cm³
3TSA = 2πr(r+h) = 2 × 22/7 × 7 × (7+10) = 44 × 17 = 748 cm²
Volume = 1540 cm³ | TSA = 748 cm²
Example 2 — Sphere: r = 6 cm (use π = 3.14)
1Volume = (4/3) × πr³ = (4/3) × 3.14 × 6³
26³ = 216 → (4/3) × 3.14 × 216
3= 4 × 3.14 × 72 = 904.3 cm³
4TSA = 4πr² = 4 × 3.14 × 36 = 452.2 cm²
Volume ≈ 904.3 cm³ | TSA ≈ 452.2 cm²
Example 3 — Cone: r = 5 cm, h = 12 cm (use π = 3.14)
1Find slant height: l = √(r² + h²) = √(25 + 144) = √169 = 13 cm
2Volume = (1/3)πr²h = (1/3) × 3.14 × 25 × 12
3= (1/3) × 942 = 314 cm³
4TSA = πr(r + l) = 3.14 × 5 × (5 + 13) = 3.14 × 5 × 18 = 282.6 cm²
Slant height = 13 cm | Volume = 314 cm³ | TSA = 282.6 cm²
Example 4 — Cuboid: l = 10, w = 4, h = 5 cm
1Volume = l × w × h = 10 × 4 × 5 = 200 cm³
2TSA = 2(lw + wh + lh) = 2(40 + 20 + 50) = 2 × 110 = 220 cm²
Volume = 200 cm³ | TSA = 220 cm²
⚠ Cone and Pyramid both have a ⅓ factor in the volume formula. TSA of cone uses slant height l — not vertical height h.
⚡ MCQ Tip: Cone and Pyramid both have ⅓ in the volume formula. Sphere TSA = 4πr² — same as 4 circles. Cylinder TSA = 2πr(r+h) — two circles plus the curved side. Slant height l = √(r²+h²).
C. π and Units
Which π to use?
Use 22/7 when radius is a multiple of 7 (easier calculation). Use 3.14 otherwise.
Exampler=7 → use 22/7 r=5 → use 3.14
Area units vs Volume units
Area is always in square units (cm²). Volume is always in cubic units (cm³).
Example1 m² = 10,000 cm² | 1 m³ = 1,000,000 cm³
Quick MCQ Revision
| Shape | Volume | TSA |
|---|---|---|
| Cube | s³ | 6s² |
| Cuboid | lwh | 2(lw+wh+lh) |
| Cylinder | πr²h | 2πr(r+h) |
| Sphere | (4/3)πr³ | 4πr² |
| Cone | (1/3)πr²h | πr(r+l) |