📦 Cube & Cuboid

• Cuboid (length $l$, breadth $b$, height $h$): volume $= lbh$; total surface area $= 2(lb+bh+hl)$ (all six faces); lateral surface area $= 2h(l+b)$ (the four walls only, no top or bottom).

• Cuboid space diagonal: the longest rod that fits inside runs corner to corner, and by Pythagoras in 3-D it is $\sqrt{l^2+b^2+h^2}$.

• Cube (side $a$): volume $= a^3$; total surface area $= 6a^2$ (six equal square faces); lateral surface area $= 4a^2$; face diagonal $= a\sqrt{2}$; space diagonal $= a\sqrt{3}$.

The single most common trap is mixing up the two diagonals: the face diagonal lies flat on one square face and is $a\sqrt{2}$, while the space diagonal cuts through the inside of the cube and is $a\sqrt{3}$. Read the question carefully before choosing which root to use.

When a volume or surface area is given and you need the side, reverse the formula with a cube root or square root. Memorise a few perfect cubes so you never have to guess: $12^3 = 1728$ and $15^3 = 3375$. A neat shortcut: if a perfect cube ends in 5, its cube root also ends in 5, so $\sqrt[3]{3375}=15$ at a glance.

Finally, the scaling rule examiners love. If two cubes (or two similar cuboids) have sides in some ratio, then the ratio of their volumes is the cube of the side ratio and the ratio of their surface areas is the square of the side ratio. So sides $2:3$ give volumes $2^3:3^3 = 8:27$ and surface areas $2^2:3^2 = 4:9$.