7.3 Shape, Dimension & Space

Why shape, dimension, and space matter on QL

The CEA QL construct on shape, dimension, and space checks whether you can read plans and solids the way first-year science, architecture, and everyday campus life require: perimeter for fencing or skirting, area for flooring or paint, volume for tanks and storerooms, and basic angle/direction language for maps and turns. Formulae may be supplied when needed, but rectangle and cuboid relationships are expected fluency — still without a calculator.

2D versus 3D conventions

A rectangle is 2D (length and width). A cuboid (rectangular box) is 3D (length, width, height). Do not multiply three lengths and call the result an area — that produces volume.

Perimeter and area of rectangles

• Perimeter P = 2(L + W) — full distance around
• Area A = L × W — interior measure

Worked example — residence common room: A rectangular room is 8 m by 5 m.

• Perimeter = 2(8 + 5) = 2 × 13 = 26 m (skirting length)
• Area = 8 × 5 = 40 m² (floor tiles)

If tiles cost R90 per m², flooring cost = 40 × 90. Compute 40 × 100 = 4 000, minus 40 × 10 = 400 → R3 600.

Worked example — unit conversion: A notice board is 120 cm by 90 cm. Area in m²?

1. Convert lengths: 120 cm = 1.2 m, 90 cm = 0.9 m
2. Area = 1.2 × 0.9 = 1.08 m²

Alternatively: area in cm² = 120 × 90 = 10 800 cm². Since 1 m² = 10 000 cm², 10 800 ÷ 10 000 = 1.08 m². Same result. Mixing 120 × 0.9 without converting both sides is a frequent trap.

Volume and surface of cuboids

• Volume V = L × W × H
• Surface area = 2(LW + LH + WH) — all six faces

Worked example — water storage cuboid: Inside dimensions 2 m by 1.5 m by 1 m.

• Volume = 2 × 1.5 × 1 = 3 m³
• 1 m³ = 1 000 litres, so capacity = 3 × 1 000 = 3 000 L

Surface area (closed tank):

• LW = 2 × 1.5 = 3
• LH = 2 × 1 = 2
• WH = 1.5 × 1 = 1.5
• Sum of opposite pairs = 3 + 2 + 1.5 = 6.5
• Surface area = 2 × 6.5 = 13 m²

Worked example — open-top planter: Same base 2 m × 1.5 m, height 1 m, but no top face.

• Bottom = 3 m²
• Sides: 2 × (2 × 1) + 2 × (1.5 × 1) = 4 + 3 = 7 m²
• Total open-top surface = 3 + 7 = 10 m²

Read whether the stem asks for capacity (volume) or material to cover (surface).

Compound rectangular shapes

Campus floor plans are often L-shapes — two rectangles joined.

Example: A lab is an L made from an 6 m × 4 m rectangle attached along the 4 m side to a 3 m × 4 m rectangle (the 4 m edges coincide so they are not double-counted as outer perimeter wrongly).

Simpler area route — add areas:

• Area₁ = 6 × 4 = 24 m²
• Area₂ = 3 × 4 = 12 m²
• Total area = 36 m²

For perimeter, walk the outer path only; do not include the internal join. Sketch and tick each outer edge once.

Angles and direction basics

Worked example: A ramp rail meets the floor at 35°. The complementary angle to reach a right angle with the vertical? Careful: complementary to 35° is 90 − 35 = 55°. Supplementary to 35° is 180 − 35 = 145°.

Compass directions (QL map language): Facing North, a 90° clockwise turn faces East; another 90° clockwise faces South; then West; then North again. A 180° turn reverses direction (North ↔ South).

Worked example — campus path: You walk 200 m East, then turn 90° left (counter-clockwise) and walk 150 m. Left from East faces North. Displacement is not 350 m in a straight line — you have a right-angled path. Straight-line distance from start uses the rectangle diagonal idea: √(200² + 150²). Without a calculator, recognise the 150–200–250 triple scaled by 50 from 3–4–5: 3-4-5 × 50 = 150-200-250. Diagonal = 250 m.

Scaling shapes

If every length of a rectangle is multiplied by k:

• Perimeter scales by k
• Area scales by k²

Example: A 3 m × 2 m poster (area 6 m²) is enlarged with k = 2 → new sides 6 m × 4 m, new area 24 m² = 6 × 2². Doubling lengths quadruples area — a classic QL trap if someone only doubles the area to 12.

For similar cuboids with length factor k, volume scales by k³. Doubling all edges multiplies volume by 8.

Common traps

Practice habit

1. Decide 1D / 2D / 3D from the question word (around, cover, hold).
2. Convert all lengths to one unit before multiplying.
3. For solids, ask whether you need inside space or outside covering.
4. For angles and maps, sketch; mark the turn clockwise or anti-clockwise explicitly.

Together with patterns and rates, these spatial skills complete the middle band of QL: relationships in number and in space, ready for data-representation items that follow.