3D Mensuration: Volume and Surface Area

Volume vs surface area

• Volume: space inside a solid (units: cm³, m³; 1 m³ = 1000 litres).
• Surface area: total area of all outer faces (units: cm², m²).

Key formulas

Solid	Volume	Total surface area
Cube (a)	a³	6a²
Cuboid (l, b, h)	l × b × h	2(lb + bh + hl)
Cylinder (r, h)	πr²h	2πr(r + h)
Cone (r, h, slant l)	⅓πr²h	πr(r + l)
Sphere (r)	(4/3)πr³	4πr²

• Curved surface area of a cylinder = 2πrh; of a cone = πrl, where l = √(r² + h²).

Examples

• Cube of side 4 cm: TSA = 6 × 16 = 96 cm².
• Cylinder r = 7 cm, h = 10 cm: volume = 22/7 × 49 × 10 = 1540 cm³.
• Sphere of radius 3 cm: volume = 4/3 × π × 27 = 36π cm³.

Exam tips (RRB NTPC)

1. A cone has one-third the volume of a cylinder with the same base and height.
2. If each side of a cube is doubled, its volume becomes 8 times.

Analogy

Volume is how much water a box can hold; surface area is how much wrapping paper it needs. A gift shop charges for paper by the outside of the box, while a water tank is valued by what fits inside. Same box, two different questions.