TACHS: Scale drawings: multiply by the scale, then convert units
Use a map or drawing scale to find real lengths, drawing lengths and real areas, and convert between centimeters, meters and kilometers.
How do you turn a length on a map or scale drawing into a real length in the right units?
Short answer
Multiply the drawing length by the scale to get the real length, or divide to go the other way. Write the unit at every step and convert at the end: 100 cm = 1 m and 1,000 m = 1 km.
Know first: Multiplying and dividing decimals, Metric units of length
Step 1 of 6
A scale is a multiplier with units
A scale of 1 cm to 40 cm means every centimeter on the drawing stands for 40 real centimeters. Multiply a drawing length by 40 to get a real length. Divide a real length by 40 to get a drawing length.
Then convert if the question asks for different units. Ask yourself which number should be bigger. Real objects are bigger than their drawings, and a length in centimeters is a bigger number than the same length in meters.
| Convert | Do this | Example |
|---|---|---|
| cm to m | Divide by 100 | 350 cm = 3.5 m |
| m to cm | Multiply by 100 | 1 m = 100 cm |
| m to km | Divide by 1,000 | 1,500 m = 1.5 km |
| km to m | Multiply by 1,000 | 3.6 km = 3,600 m |
Smaller unit to larger unit: the number gets smaller. Larger to smaller: the number gets bigger.
Step 2 of 6
Scale first, units second
Worked example: A kitchen counter
A kitchen plan uses the scale 1 cm to 40 cm. A counter is 4.5 cm long on the plan. How long is the real counter, in meters?
- A0.18
- B1.8
- C18
- D180
- 1
Apply the scale
4.5 × 40 = 180 cm.
- 2
Convert
180 cm ÷ 100 = 1.8 m.
- 3
Sense check
1.8 meters is a reasonable length for a kitchen counter.
- 4
See why the other choices are there
180 is the length in centimeters. 18 and 0.18 move the decimal point the wrong number of places.
Answer
The counter is 1.8 meters long.
Wrong: A plan uses 1 cm to 2 m. A room is 5 cm by 3 cm, so the plan area is 15, and the real area is 15 × 2 = 30 square meters.
Right: Real sides: 5 × 2 = 10 m and 3 × 2 = 6 m. Real area: 10 × 6 = 60 square meters.
Both sides are doubled, so the area grows by 2 × 2 = 4 times, not 2 times. Convert each side to its real length before you multiply.
Step 3 of 6
Going the other way, and finding a scale
To find a drawing length, convert the real length to the scale's units, then divide. A map uses 1 cm to 400 m, and a trail is 3.6 km long. Convert: 3.6 km = 3,600 m. Divide: 3,600 ÷ 400 = 9 cm on the map.
To find a scale, divide the real length by the drawing length. If a 15-meter building is 6 cm tall on a drawing, each centimeter stands for 15 ÷ 6 = 2.5 meters.
Step 4 of 6
Scale, then units, then a sense check
Check yourself · Question 1
A floor plan uses the scale 1 cm to 60 cm. A bookcase is 3.5 cm wide on the plan. How wide is the real bookcase, in meters?
Answer: B
Real width: 3.5 × 60 = 210 cm. Convert: 210 ÷ 100 = 2.1 m.
Trap answer: 210
Right number, wrong unit
- Why it looks right
- 210 comes straight out of the scale, and the work feels finished.
- Why it's wrong
- The scale gives centimeters because the scale is written in centimeters. The question asks for meters, so one more step is needed.
- The right answer
- 2.1 m, because 210 cm ÷ 100 = 2.1 m.
Check yourself · Question 2
A map uses the scale 1 cm to 300 m. A road is 4.2 km long. How long is the road on the map, in centimeters?
Answer: C
Convert the real length: 4.2 km = 4,200 m. Divide by the scale: 4,200 ÷ 300 = 14 cm.
Check yourself · Question 3
A drawing uses the scale 1 cm to 4 m. A patio measures 3 cm by 2.5 cm on the drawing. What is the real area of the patio, in square meters?
Answer: D
Real sides: 3 × 4 = 12 m and 2.5 × 4 = 10 m. Area: 12 × 10 = 120 square meters.
Common mistake
"To change 180 cm to meters, I multiply by 100."
- Why it's tempting
- There are 100 centimeters in a meter, so 100 belongs in the problem somewhere.
- Do this instead
- Meters are bigger units, so you need fewer of them. Divide by 100: 180 cm = 1.8 m. Ask which way the number should move before you calculate.
Step 5 of 6
What to remember
Remember
- Multiply by the scale to get real lengths, and divide to get drawing lengths.
- Convert units at the end: 100 cm = 1 m and 1,000 m = 1 km, checking whether the number should grow or shrink.
- For area, convert each side to its real length first, then multiply.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, what is the real length in centimeters before you convert, and which wrong choice is that number?