Solids

Volume = a³, Total surface area = 6a², Lateral surface area = 4a², Diagonal = a√3

For a cube: Volume = a³, TSA = 6a² (6 faces), LSA = 4a² (4 side faces), Diagonal = a√3. Common mistake: Face diagonal is a√2, but space diagonal is a√3.

Length, Breadth, Height

Volume of cuboid = length × breadth × height. You need all three dimensions. Diagonal and surface area can be calculated from these three dimensions but are not needed for volume.

27 cm³

TSA = 6a² = 54, so a² = 9, hence a = 3 cm. Volume = a³ = 3³ = 27 cm³. Step-by-step: 54 ÷ 6 = 9, √9 = 3, 3³ = 27.

160 cm²

Lateral surface area = 2(l + b) × h = 2(6 + 4) × 8 = 2 × 10 × 8 = 160 cm². This covers the 4 vertical faces only, not the top and bottom.