3D Mensuration • Topic 1 of 4

Cube & Cuboid

A cube has all sides equal (side a): volume a³, total surface area 6a², and space diagonal a√3 — the longest straight line that fits inside, useful for "will a rod fit" questions. A cuboid (l, b, h) has volume lbh, TSA 2(lb + bh + hl), and diagonal √(l²+b²+h²). Two CAT staples live here. First, the painted-cube problem: a large cube of side n smaller cubes is painted and cut — corner cubes (3 faces) always number 8, edge cubes (2 faces) are 12(n−2), face-centre cubes (1 face) are 6(n−2)², and fully unpainted interior cubes are (n−2)³. Second, scaling: double every edge and volume becomes 8× while surface area becomes 4×. Always check whether a box is open (5 faces) or closed (6 faces) before applying the TSA formula — this single distinction trips up most students.

✅ Solved examples

1. The space diagonal of a cube is 12√3 cm. Find its volume.
Diagonal = a√3 = 12√3 ⇒ a = 12. V = a³ = 12³ = 1728 cm³.
2. A cuboidal tank is 6 m × 4 m × 3 m. Find its volume and total surface area.
V = 6×4×3 = 72 m³. TSA = 2(6×4 + 4×3 + 3×6) = 2(24+12+18) = 2×54 = 108 m².
3. A cube of side 4 cm is painted on all faces and cut into 1 cm cubes. How many small cubes have exactly two faces painted?
n = 4. Two-face (edge) cubes = 12(n−2) = 12×2 = 24.
4. Each edge of a cube is increased by 50%. By what percent does its surface area increase?
TSA scales as a². New side 1.5a ⇒ area factor 1.5² = 2.25 ⇒ increase of 125%.

✏️ Practice — try these, take hints as needed

1. Find the longest rod that fits inside a room 12 m × 9 m × 8 m.
Longest line = space diagonal.
√(l²+b²+h²).
√(144+81+64) = √289.
17 m
2. The surface area of a cube is 150 cm². Find its volume.
6a² = 150.
a² = 25 ⇒ a = 5.
V = a³.
125 cm³
3. A 5×5×5 cube is painted and cut into unit cubes. How many have no face painted?
Interior cube (n−2)³.
n = 5.
3³.
27
4. A cuboid has dimensions in ratio 3 : 2 : 1 and volume 162 cm³. Find its longest edge.
Let edges 3x, 2x, x.
6x³ = 162 ⇒ x³ = 27.
x = 3, longest = 3x.
9 cm
5. If each edge of a cube is doubled, the volume increases by how many times the original?
V ∝ a³.
2³ = 8.
New = 8× old.
8 times (an increase of 7×)

📝 Topic test — 8 questions

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