A plan is drawn to one scale. Someone measures it with a ruler graduated to a different scale. Every reading they take is wrong — but predictably wrong, and therefore correctable.
This post covers the two correction formulas, explains why the area formula is squared, and works through a complete numerical example.
The Situation
If a wrong measuring scale is used to measure the length of a line already drawn on a plan, the measured length will not be correct.
Note carefully what has and has not gone wrong here.
The plan itself is perfectly correct. It was drawn accurately to its intended scale. The error lies entirely in the instrument used to read it — a scale rule graduated for a different ratio.
This is what makes the correction possible. Since the drawing is right and only the reading is wrong, applying the ratio between the two scales recovers the true value exactly.
The Correct Length Formula
Correct length = (R.F. of wrong scale / R.F. of correct scale) × Measured length
Getting the Ratio the Right Way Up
This is where marks are lost, so it is worth thinking through rather than memorising.
The wrong scale is on top and the correct scale is underneath.
Here is a way to check yourself. Suppose the wrong scale used was a larger scale than the correct one — meaning it draws ground bigger, so it reads distances as smaller than they really are. To correct this you must increase the measured value. And indeed, with the larger R.F. on top, the ratio comes out greater than 1, which increases the answer. The formula behaves correctly.
The Correct Area Formula
Correct area = (R.F. of wrong scale / R.F. of correct scale)2 × Measured area
As the area is the product of two distances, the ratio is squared.
Why the Square Appears
An area has two dimensions. If every length reading is out by a certain ratio, then both the length and the breadth of any shape are out by that same ratio — so the area is out by the ratio multiplied by itself.
Consider a rectangle. Its area is length × breadth. If each is wrong by a factor of 2, the area is wrong by 2 × 2 = 4, not 2.
This is exactly the same reasoning as the SF2 in shrinkage problems, and both come from the same fact — area is a product of two lengths.
Solved Example
Problem: A surveyor measured the distance between two points marked on a plan drawn to a scale with R.F. = 1:100, and found it to be 50 m. Later he detected that he had used a wrong scale of R.F. = 1:50 for the measurement.
Determine the correct length. What would be the correct area if the measured area is 60 m2?
Step 1 — Identify Which Is Which
| R.F. | |
|---|---|
| Wrong scale (the one used for measuring) | 1/50 |
| Correct scale (the one the plan was drawn to) | 1/100 |
Sorting this out first is half the battle. The plan was drawn at 1:100 — that is the correct scale. The measurement was taken with a 1:50 rule — that is the wrong scale.
Step 2 — Correct Length
Correct length = (R.F. of wrong scale / R.F. of correct scale) × measured length
= (1/50) ÷ (1/100) × 50
= (100/50) × 50
= 2 × 50
= 100 m
Step 3 — Correct Area
Correct area = [(1/50) ÷ (1/100)]2 × measured area
= (2)2 × 60
= 4 × 60
= 240 m2
Step 4 — Sanity Check
Always test whether the answer is reasonable.
- The length doubled, from 50 m to 100 m — consistent with a ratio of 2.
- The area quadrupled, from 60 m2 to 240 m2 — consistent with a ratio of 22 = 4.
If your area answer is only doubled, you have forgotten to square the ratio. That check takes two seconds and catches the most common error in the topic.
The Complete Procedure
- Identify the correct scale — the one the plan was drawn to.
- Identify the wrong scale — the one used for measuring.
- Form the ratio with the wrong scale R.F. on top.
- For a length, multiply the measured value by that ratio.
- For an area, multiply by the square of the ratio.
- Check that the area factor is the square of the length factor.
Wrong Scale versus Shrinkage — Two Different Problems
These two topics use very similar mathematics, which makes them easy to confuse.
| Wrong Measuring Scale | Map Shrinkage | |
|---|---|---|
| What went wrong | The instrument used to read the plan | The paper itself changed size |
| Is the plan still correct? | Yes — only the reading is wrong | No — the drawing has physically changed |
| Length correction | × (wrong R.F. / correct R.F.) | ÷ SF |
| Area correction | × (wrong R.F. / correct R.F.)2 | ÷ SF2 |
Both square the factor for areas, and for the same reason. What differs is whether the ratio is multiplied or divided — and that follows from whether the drawing or the instrument was at fault.
Formula Summary
| Quantity | Expression |
|---|---|
| Correct length | (R.F. of wrong scale / R.F. of correct scale) × measured length |
| Correct area | (R.F. of wrong scale / R.F. of correct scale)2 × measured area |
| Reason for the square | Area is the product of two distances |
Quick Revision Notes
- Using a wrong measuring scale on a correctly drawn plan gives an incorrect measured length.
- The plan is not at fault — only the instrument used to read it.
- Correct length = (R.F. of wrong scale / R.F. of correct scale) × measured length.
- Correct area = (R.F. of wrong scale / R.F. of correct scale)2 × measured area.
- The ratio is squared for area because area is the product of two distances.
- The wrong scale R.F. goes on top of the ratio.
- In the standard example, R.F. 1/50 used on a 1/100 plan gives a ratio of 2, so 50 m becomes 100 m and 60 m2 becomes 240 m2.
- Always check that the area factor is the square of the length factor.
Mistakes Students Commonly Make
- Forgetting to square the ratio for area. This is the single most frequent error.
- Inverting the ratio. The wrong scale R.F. is in the numerator.
- Mixing up which scale is which. The correct scale is the one the plan was drawn to; the wrong scale is the one used for measuring.
- Working with the denominators instead of the fractions. Use the R.F. values as fractions, so 1/50 ÷ 1/100 = 2, not 50/100 = 0.5.
- Confusing this with shrinkage. Here you multiply by the ratio; in shrinkage you divide by SF.
- Skipping the sanity check that the area factor equals the length factor squared.
Conclusion
When a plan is measured with the wrong scale rule, the drawing remains correct and only the reading is at fault — which makes the error fully recoverable. Multiply the measured length by the ratio of wrong R.F. to correct R.F., and multiply the measured area by the square of that ratio. The square appears because an area is two lengths multiplied together, so any error in length affects it twice over. Get the ratio the right way up, remember the square, and check that the area factor is the length factor squared.
Frequently Asked Questions
What happens if a wrong measuring scale is used?
The length measured from an already drawn plan will not be correct, even though the plan itself is accurate. The error lies in the measuring scale, not the drawing.
What is the formula for correct length?
Correct length = (R.F. of wrong scale / R.F. of correct scale) × measured length.
What is the formula for correct area?
Correct area = (R.F. of wrong scale / R.F. of correct scale)² × measured area.
Why is the ratio squared for area?
Because area is the product of two distances. If every length is wrong by a certain ratio, both dimensions of the area are wrong by that ratio, so the area is wrong by the ratio squared.
Which R.F. goes in the numerator?
The R.F. of the wrong scale — the one actually used for measurement.
In the standard example, what is the correct length?
With a plan at R.F. 1/100 measured using a 1/50 scale, the ratio is 2, so a measured 50 m corrects to 100 m.
And the correct area?
The area ratio is 2² = 4, so a measured 60 m² corrects to 240 m².
How does this differ from a shrinkage problem?
In a wrong scale problem the plan is correct and only the instrument is at fault, so the measured value is multiplied by the ratio. In a shrinkage problem the paper itself has changed, so the measured value is divided by the shrinkage factor. Both square the factor for areas.
What is a quick way to check the answer?
Confirm that the factor applied to the area is the square of the factor applied to the length. If the length doubled, the area should quadruple.