Working with scale and ratio in drawings and models, reading and using scales (for example 1:2, 1:5, 1:10 and enlargement scales), calculating areas and volumes for material estimation, using trigonometry and geometry to find lengths and angles, surface area and capacity calculations, and converting between units in a design context.

What this dot point is asking

Edexcel wants you to work with scale and ratio in drawings and models, read and use scales, calculate areas and volumes for material estimation, use trigonometry and geometry to find lengths and angles, work out surface area and capacity, and convert between units in a design context.

The answer

Scale and ratio in drawings

Ratio, proportion and scaling models

Ratio sets the relative sizes of parts (a $2 \colon 1$ ratio means one part is twice another) and is used in proportion and in scaling a whole model up or down by a single factor. Note that scaling a length by a factor $k$ scales area by $k^2$ and volume by $k^3$, which matters for material and capacity.

Geometry and trigonometry

Area, surface area, volume and capacity

Area estimates how much sheet material to buy and the surface area to finish (paint, coat); volume estimates the material mass (volume times density) and the capacity of containers. Convert to consistent units before calculating.

Converting units

Work in one set of units: $10$ mm $= 1$ cm, $100$ cm $= 1$ m, $1000$ mm $= 1$ m. For area and volume, the conversion factor is squared or cubed ($1$ cm squared $= 100$ mm squared; $1$ cm cubed $= 1000$ mm cubed). Getting units consistent first prevents most errors.

Examples in context

An architect or product designer draws a large item at $1 \colon 10$ or $1 \colon 100$ to fit the page and a tiny electronic part at $5 \colon 1$ to detail it, reading real sizes off the scale. Material is estimated by area (sheet needed, surface to finish) and by volume (mass from volume times density, and the capacity of bottles and tanks in litres). Trigonometry sizes braces, slopes and frames. Scaling a prototype up reminds the designer that doubling the length multiplies the volume and material eightfold. Using scale and ratio correctly, calculating area, volume and capacity, and converting units consistently, are the maths-in-context skills Edexcel tests most often here.

Try this

Q1. A part is drawn at $1 \colon 2$ and measures $60$ mm on the drawing. What is its real length? [1 mark]

• Cue. Real length $= 60 \times 2 = 120$ mm.

Q2. A rectangular sheet is $1.2$ m by $0.8$ m. Find its area in square metres. [1 mark]

• Cue. $A = 1.2 \times 0.8 = 0.96$ square metres.

Q3. A cylindrical bottle has radius $3$ cm and height $20$ cm. Find its volume and its capacity in litres. [2 marks]

• Cue. $V = \pi r^2 h = \pi \times 9 \times 20 = 180\pi = 565$ cubic centimetres; capacity $= 565 \div 1000 = 0.57$ litres.