Surface Area and Volume Formulas for Class 10 (All Shapes)

CUBE (side = a)

• TSA = 6a²
• LSA (4 faces) = 4a²
• Volume = a³
• Diagonal = a√3
• Face diagonal = a√2

CUBOID (length = l, breadth = b, height = h)

• TSA = 2(lb + bh + hl)
• LSA = 2h(l + b)
• Volume = l × b × h
• Diagonal = √(l² + b² + h²)

Note: For a cube, all three dimensions are equal (l = b = h = a), so TSA = 2(a² + a² + a²) = 6a².

CYLINDER (radius = r, height = h)

• TSA = 2πr(r + h)
• CSA = 2πrh
• Volume = πr²h

CONE (radius = r, height = h, slant height = l)

• TSA = πr(r + l)
• CSA = πrl
• Volume = (1/3)πr²h
• Slant height: l = √(r² + h²)

Relationship: l² = r² + h² (Pythagoras theorem applied to cone)

Remember: The slant height l is the distance from the apex (tip) to the base edge along the surface, not through the interior.

SPHERE (radius = r)

• TSA = 4πr²
• CSA = 4πr² (sphere has only one surface)
• Volume = (4/3)πr³

Note: A sphere has no distinct CSA and TSA — both are 4πr².

HEMISPHERE (radius = r)

• TSA = 3πr² (curved surface + circular base)
• CSA = 2πr² (only the curved part)
• Volume = (2/3)πr³

Explanation:

• Hemisphere CSA = half of sphere TSA = (1/2)(4πr²) = 2πr²
• Hemisphere TSA = CSA + area of circular base = 2πr² + πr² = 3πr²
• Hemisphere Volume = half of sphere volume = (1/2)(4/3)πr³ = (2/3)πr³

FRUSTUM (a cone with its top cut off) Given: radii R (bigger base) and r (smaller base), height h, slant height l

• TSA = π[R² + r² + l(R + r)]
• CSA = π(R + r)l
• Volume = (πh/3)(R² + r² + Rr)
• Slant height: l = √[h² + (R − r)²]

Explanation:

• The frustum has two circular faces (area πR² and πr²) plus the curved lateral surface.
• CSA = π(R + r)l is derived from the difference of two cone CSAs.
• Volume = (πh/3)(R² + Rr + r²) comes from subtracting the smaller cone from the larger cone.

Note: When r = 0 (tip not cut), frustum becomes a full cone. When r = R, it becomes a cylinder.

Useful values: π ≈ 3.14159, π ≈ 22/7 (for approximate calculations)

Slant height formulas:

• Cone: l = √(r² + h²)
• Frustum: l = √[h² + (R−r)²]