7.3 Shape, Dimension and Space
Key Takeaways
- QL geometry is measurement in a campus or laboratory context, not MAT Euclidean proof
- 2D work uses length and area units; 3D work adds height and volume; angles use degrees and compass direction
- Perimeter of a rectangle is 2(l + w); area is l × w; volume of a cuboid is l × w × h, with the formula given when needed
- Radius is centre to rim; diameter is twice the radius; circumference and area formulae appear in the item
- A scale diagram converts map length to actual length by multiplying by the stated scale, then keeping real-world units
Measurement in context, not proof
The QL subdomain Shape, dimension and space asks you to use conventions for measuring and describing 2-dimensional and 3-dimensional objects, angles, and direction, and to compute perimeters, areas, and volumes of simple shapes such as rectangles and cuboids. Test Content also lists apply properties of simple geometric shapes to determine measurements. QL is still 50 multiple-choice questions, still calculator-free, and still not MAT. Independent OpenExamPrep space practice uses noticeboards, storage crates, circular garden beds, and campus maps. It does not copy confidential flask-and-grid booklet diagrams, and it does not claim official sponsorship.
MAT may ask you to prove two triangles congruent or to quote a circle theorem. A QL item gives the formula you need and asks whether the noticeboard border, the crate capacity, or the map path is the number in the options. If you start a two-column Euclidean proof, you have walked into the wrong paper.
Conventions: what the number is allowed to mean
| Object | Typical measurements | SI-style units in QL stories |
|---|---|---|
| Rectangle, noticeboard, field | length, width, perimeter, area | m, cm; m², cm² |
| Cuboid, crate, room | length, width, height, volume, sometimes surface area | m³, cm³, litres after a stated conversion |
| Circle, round table, fountain | radius, diameter, circumference, area | m, m²; formula given |
| Plan or map | scale, bearing or compass direction | cm on paper to m on campus |
| Angle | degrees; right angle 90°, straight 180°, full turn 360° | ° |
A 2D quantity has no thickness you are asked to use. Painting a noticeboard is area. Taping its edge is perimeter. A 3D crate has a height; filling it is volume. Using cm² for a volume, or adding length to area, is a unit trap, not a small slip.
Direction. From north, 90° clockwise is east, 180° is south, 270° is west, 360° returns to north. A campus map that says the clinic lies 90° clockwise from north of the library is saying the clinic is east of the library. “90°” without a starting direction is not yet a bearing you can walk.
Radius versus diameter. The radius r runs from the centre to the rim. The diameter d runs from rim to rim through the centre, so d = 2r and r = d / 2. Using the diameter in a formula that asked for radius doubles a length and can quadruple an area, because area grows with r².
Rectangles: perimeter and area
Formulae, when the item gives them: perimeter P = 2(l + w) and area A = l × w.
Worked example: noticeboard. A faculty board is 1.2 m by 0.8 m.
l + w = 1.2 + 0.8 = 2.0 m. P = 2 × 2.0 = 4.0 m of border tape. A = 1.2 × 0.8. 12 × 8 = 96, so 1.2 × 0.8 = 0.96 m² of pin space.
Traps: 0.96 m (area with a length unit), 2.0 m (l + w once, which is only half the perimeter), 1.92 m (twice 0.96, mixing area with a length). Border tape is 4.0 m. Covering the board is 0.96 m². Those are different jobs.
Worked example: sports-field strip. A practice rectangle is 40 m by 25 m. P = 2(40 + 25) = 2 × 65 = 130 m. A = 40 × 25 = 1 000 m². If marking paint covers 1 000 m², that is the interior, not the 130 m boundary. If a rope follows the boundary, buy 130 m, not 1 000 m.
Cuboids: volume, and packing
Formula given: volume V = l × w × h.
Worked example: sample crate. Interior 40 cm by 25 cm by 20 cm. 40 × 25 = 1 000; 1 000 × 20 = 20 000 cm³. If the item states 1 000 cm³ = 1 L, the crate holds 20 L. Treating 20 000 cm³ as 20 000 L inflates the capacity a thousand times.
How many 5 cm cubes pack the crate if they sit flush? 40 / 5 = 8 along the length. 25 / 5 = 5 along the width. 20 / 5 = 4 along the height. 8 × 5 × 4 = 160 cubes. Adding 8 + 5 + 4 = 17 counts edges, not packed cubes. Using only the 40 × 25 face gives 8 × 5 = 40 cubes and forgets the height.
Surface area, if the formula S = 2(lw + lh + wh) is given: lw = 1 000, lh = 800, wh = 500; sum 2 300; S = 4 600 cm². That is wrapping paper, not capacity. Do not report 4 600 as the volume.
A floor plan of the crate is a 40 cm by 25 cm rectangle. It is a 2D representation of a 3D object. You cannot read volume from the plan until the stem gives the missing height.
Circles: diameter first, then the given formula
Worked example: garden bed. A circular bed has diameter 7 m. Radius r = 7 / 2 = 3.5 m. The item gives π = 22/7 and circumference C = πd. C = (22/7) × 7 = 22 m of edging. The item also gives area A = πr². r² = (7/2)² = 49/4. A = (22/7) × (49/4) = (22 × 7) / 4 = 154 / 4 = 38.5 m² of soil surface.
Using 7 m as the radius would compute A = (22/7) × 49 = 154 m², four times too large, because (2r)² = 4r². Using 3.5 m as the diameter would halve the circumference. Name r or d before you substitute.
Scale diagrams
Worked example: campus map. Scale 1 cm : 20 m. A path measures 7.5 cm on the map. Actual length: 7.5 × 20. 7 × 20 = 140; 0.5 × 20 = 10; total 150 m. A building 40 m long is 40 / 20 = 2 cm on the map.
Traps: 7.5 m (map centimetres read as metres), 20 m (the scale quoted once), 150 cm (the product kept in map units). Scale is a ratio of lengths. It does not convert a map centimetre into a campus square metre unless the item asks for area and you square the scale factor.
If a scale diagram of a round fountain shows diameter 4 cm at 1 cm : 2 m, actual diameter = 8 m, actual radius = 4 m. Then apply the circle formula the item prints, using 4 m, not 4 cm.
Angles you can walk
A right-angled turn on a path is 90°. Two successive right turns heading north, then east, then south, have changed direction by 180° from the original heading. A rectangular field has four right angles; the walk around it is a 360° turn in four 90° pieces, matching perimeter as a closed loop, not as a proof that the figure “is a rectangle by theorem.”
A method
- Decide 2D or 3D, then perimeter, area, or volume.
- Write units on the line: m versus m² versus m³; cm³ versus litres after the given conversion.
- For a circle, mark radius or diameter before substituting into the given formula.
- For a map, multiply by the scale and switch to campus units.
- For direction, start from the stated compass point, then apply the stated degrees.
Traps
- Reporting 0.96 m² as 0.96 m of tape
- Adding 8 + 5 + 4 cubes instead of multiplying the packing counts
- Using diameter in a radius formula
- Leaving a scale product in centimetres
- Treating a floor plan as a volume
- Writing a MAT congruence proof when the item asked for tape length
Independent OpenExamPrep QL space teaching is measurement that still names the board, the crate, the bed, or the path after the last product.
A faculty noticeboard is 1.2 m by 0.8 m. The length of border tape for the perimeter is:
A sample crate measures 40 cm by 25 cm by 20 cm. Its interior volume is:
A campus map uses the scale 1 cm : 20 m. A path measures 7.5 cm on the map. The actual path length is: