R2 - Scale diagrams and maps

R2 - Scale diagrams and maps

A scale drawing shrinks a real place while keeping its lengths in the same proportions. In R2 you will turn map lengths into real distances, work backwards to drawing lengths, and choose dimensions for a scale plan. You will then use a plan to solve a geometrical problem.

Reading and reversing a scale

A rectangular garden appears as 7 cm7\,\mathrm{cm} by 4 cm4\,\mathrm{cm} on a plan with scale 1:5001:500. There is a 4 m4\,\mathrm{m} gate in its boundary. How much fence is needed around the rest? A little drawing can answer a much larger question.

You already know

Ratio order matters. Make units agree before simplifying a ratio. A kilometre is 1000 m1000\,\mathrm{m}, and a metre is 100 cm100\,\mathrm{cm}.

A scale of 1:5001:500 means one length unit on the drawing represents 500500 of the same unit in reality. So 1 cm1\,\mathrm{cm} represents 500 cm=5 m500\,\mathrm{cm}=5\,\mathrm{m}. It does not mean 1 cm1\,\mathrm{cm} represents 500 m500\,\mathrm{m}.

Scale factor

The multiplier from one set of corresponding lengths to another. For a plan with scale drawing : actual =1:n=1:n, multiply drawing lengths by nn to get actual lengths. Multiply actual lengths by 1n\frac1n to get drawing lengths.

The smaller scale factor 1n\frac1n makes a plan. The reverse factor nn recovers reality. Angles and shape stay the same when every length is scaled equally.

Scale drawing lengths

For a scale 1:n1:n, where n>0n>0, using the same units on both sides,

actual length=drawing length×ndrawing length=actual length÷n\begin{aligned} \text{actual length}&=\text{drawing length}\times n\\ \text{drawing length}&=\text{actual length}\div n \end{aligned}

Not on the formulae sheet: learn it.

Write a unit at the intermediate stage. It prevents a centimetre answer being mistaken for metres.

Worked example 1: map to reality

Question. A map has scale 1:25 0001:25\,000. Two places are 4.6 cm4.6\,\mathrm{cm} apart on the map. Work out their actual distance in kilometres. Non-calculator.

Plan. Scale up in centimetres, then convert to kilometres.

4.6×25 000=115 000actual centimetres115 000÷100=1150actual metres1150÷1000=1.15actual kilometres\begin{aligned} 4.6\times25\,000&=115\,000 &&\text{actual centimetres}\\ 115\,000\div100&=1150 &&\text{actual metres}\\ 1150\div1000&=1.15 &&\text{actual kilometres} \end{aligned}

Why keep centimetres first? The scale ratio compares equal units.

Answer. 1.15 km1.15\,\mathrm{km}.

Check. Each map centimetre is 0.25 km0.25\,\mathrm{km}; 4.6×0.25=1.154.6\times0.25=1.15.

Reverse the direction while keeping the same map scale. Reality must be converted into centimetres before dividing.

Worked example 2: reality to map

Question. The same map has scale 1:25 0001:25\,000. A straight road is 1.8 km1.8\,\mathrm{km} long. Work out its length on the map. Non-calculator.

Plan. Convert the real distance to centimetres, then divide by 25 00025\,000.

1.8×1000×100=180 000actual centimetres180 000÷25 000=7.2map centimetres\begin{aligned} 1.8\times1000\times100&=180\,000 &&\text{actual centimetres}\\ 180\,000\div25\,000&=7.2 &&\text{map centimetres} \end{aligned}

Why divide? The drawing is smaller than the real road.

Answer. 7.2 cm7.2\,\mathrm{cm}.

Check. 7.2×0.25=1.8 km7.2\times0.25=1.8\,\mathrm{km}, using the centimetre-to-kilometre check.

A map's printed ratio depends on its printed size. Double every printed length and a 1:25 0001:25\,000 map becomes 1:12 5001:12\,500. A scale bar enlarged along with the map still works, because the bar changes size with the distances.

Watch out

Wrong: 1.8÷25 000=0.000072 cm1.8\div25\,000=0.000072\,\mathrm{cm}. This mixes kilometres with centimetres. Right: convert 1.8 km1.8\,\mathrm{km} to 180 000 cm180\,000\,\mathrm{cm} before dividing.

When measuring a paper map, measure the stated route, not a different straight-line distance. The next road length is already supplied, so no ruler is needed.

Making a scale plan

A useful scale makes the whole drawing fit while keeping enough detail to measure.

Suppose 1 cm1\,\mathrm{cm} represents 2 m2\,\mathrm{m}. Since 2 m=200 cm2\,\mathrm{m}=200\,\mathrm{cm}, the ratio scale is 1:2001:200. Apply that one scale to every length, draw straight edges with a ruler, and label the scale. A drawing stretched in only one direction is not a scale plan.

Worked example 3: a plan at one centimetre per metre

Question. A rectangular room is 8 m8\,\mathrm{m} long and 5 m5\,\mathrm{m} wide. Work out the drawing dimensions for a plan where 1 cm1\,\mathrm{cm} represents 1 m1\,\mathrm{m}. Non-calculator.

Plan. Divide each real dimension by the metres represented by one drawing centimetre.

8÷1=8drawing length in centimetres5÷1=5drawing width in centimetres\begin{aligned} 8\div1&=8 &&\text{drawing length in centimetres}\\ 5\div1&=5 &&\text{drawing width in centimetres} \end{aligned}

Why use the same divisor? A scale plan must preserve the room's shape.

Answer. Draw a rectangle 8 cm8\,\mathrm{cm} by 5 cm5\,\mathrm{cm}, with right angles. Label the scale 1:1001:100.

Check. 8:58:5 is the length-to-width ratio in both the room and the plan.

Change only the scale: one centimetre now represents twice as much real length. Both drawing dimensions must halve.

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