Cubes and Cuboids

Module 06 — Lesson 0026 · Mensuration 3D · In Depth · 40 min

3D mensuration's two primitives: cube (all faces squares) and cuboid (box). Volume is space occupied, surface is skin. Scaling, cutting/painting, and melting are just counting how volume redistributes.

Cube (a) vs cuboid (l,b,h) — same count, different regularity.

1. The Cube — Side `a`

A cube has 6 equal square faces, 12 equal edges, 8 vertices.

Property	Formula	What it means
Volume	`a³`	Space occupied
Total Surface Area (TSA)	`6a²`	All 6 faces
Lateral Surface Area (LSA)	`4a²`	4 side walls (no top/bottom)
Face diagonal	`a√2`	Diagonal on one square face
Space diagonal	`a√3`	Longest inside — corner to opposite corner

LSA = TSA − 2×base = 6a² −2a² =4a². Space diagonal via Pythagoras twice: face diagonal a√2, then √(a²+(a√2)²)=a√3.

2. The Cuboid — `l, b, h`

Box with 6 rectangles — opposite faces identical (lb, bh, hl pairs).

Property	Formula
Volume	`l × b × h`
TSA	`2(lb + bh + hl)`
LSA	`2h(l + b)` — perimeter of base × height
Space diagonal	`√(l² + b² + h²)`

3. Advanced Concepts & Logic

a) Scaling (Changing Dimensions by Factor `k`)

If edge of cube is multiplied by k (e.g., doubled → k=2):

Surface area scales by k² (×4 for k=2)
Volume scales by k³ (×8 for k=2)

1D scales as k, 2D as k², 3D as k³ — same as Module 08 geometry ratios.

Scaling: 1D ×k, 2D ×k², 3D ×k³.

b) Percentage Change

If side increases by x%: volume increase = (1+x/100)³ −1. Use successive formula: first x%→ x + x + x²/100 for area, then combine again for volume.

Worked: Side +20% → (1.20)³=1.728 → +72.8% volume. Surface: 1.20²=1.44 → +44%.

c) Cutting and Painting — The N-Cube

Large cube side N (in units of small cube side 1) cut into small cubes of side 1:

Total small cubes = N³
3 faces painted: 8 (corners — always 8)
2 faces painted: (N−2) × 12 — edges (12 edges × (N−2) middle pieces)
1 face painted: (N−2)² × 6 — face centers
0 faces painted: (N−2)³ — inner core (only if N>2)

Cutting: corners always 8; edges, faces, core scale with (N−2).

d) Melting and Recasting

Shape melted → new shape: Volume(old) = Volume(new), surface area usually changes. Example: sphere melted into cube → 4/3 πr³ = a³.

e) Fitting Largest Shapes

Largest sphere in a cube (side a): diameter = a → radius = a/2 (sphere touches all faces).
Largest cube in a sphere (radius R): space diagonal of cube = diameter of sphere → a√3 = 2R → a = 2R/√3.

Summary of Variables

Symbol	Meaning
`a`	Side of cube
`l, b, h`	Length, breadth, height of cuboid
`d`	Diagonal
`V`	Volume
`SA`	Surface area

Volume of cube side a?

TSA of cube side a?

Space diagonal of cube side a?

Edge doubled (k=2) → volume becomes?

Side +20% → volume increase?

Large cube N=3 cut into 1-unit cubes → 3 faces painted?

Largest sphere in cube side a → radius?

Melting: sphere to cube → what is equal?

Notes

Cube (a) :-

6 faces, 12 edges, 8 vertices ; Vol ⇒ a³ ; TSA 6a² ; LSA 4a²

Face diag a√2 ; Space diag a√3

Cuboid (l,b,h) :-

Vol ⇒ lbh ; TSA 2(lb+bh+hl) ; LSA 2h(l+b) ; Space diag √(l²+b²+h²)

Scaling & Cutting :-

Edge ×k → SA ×k², Vol ×k³ ; +x% → Vol (1+x/100)³−1 (20%→72.8%)

Cut N³ small cubes: 3F→8, 2F→(N−2)×12, 1F→(N−2)²×6, 0F→(N−2)³

Melt → vol equal ; Largest sphere in cube: D=a ; cube in sphere: a√3=2R

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