---
title: "Scale and area on topo sheet"
subject: "Surveying I"
chapter: 1
key: "semester_3/surveying_i:scale-plot-18000-sqm"
source: https://www.bctengineer.com/q/semester_3/surveying_i/scale-plot-18000-sqm
site: BCT Engineering
---

# Scale and area on topo sheet

Surveying I · Chapter 1: Introduction

- Asked in: 2072 Chaitra, 6 marks
- Video: [Geography Mapwork: How to calculate Area on a map](https://www.youtube.com/watch?v=p17qym2sIso) — Everything Geography by Thapelo
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/surveying_i/scale-plot-18000-sqm
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-1

## Question

A plan represents an area of 18000 m² and measures 8 cm × 9 cm. Find the scale of the plot and indicate through a sketch how a suitable scale can be constructed to read up to 1 m in the plan. If the same plan is to be drawn on a topo sheet with a scale of 1:12500, what will be the represented area of that plan on the sheet?

## Short answer

RF ≈ 1/1581 (1 cm = 15.81 m). A diagonal scale 9.49 cm long reads 1 m up to 150 m. On a 1:12500 sheet the plan area is 1.152 cm².

## Answer

#### Scale of the plan
Map area $= 8 \times 9 = 72$ cm$^2$. Ground area $= 18000 \times 10^{4} = 1.8 \times 10^{8}$ cm$^2$.

$$
\text{RF}^2 = \frac{72}{1.8 \times 10^{8}} = 4.0 \times 10^{-7}
$$

$$
\text{RF} = 6.3246 \times 10^{-4} = \frac{1}{1581.1}
$$

So 1 cm on the plan $= 15.81$ m on the ground (approx. 1:1581).

#### Sketch of a scale reading 1 m
Take a maximum reading of 150 m: length $= 150 \times 100/1581.1 = 9.49$ cm (100 m $= 6.32$ cm, 10 m $= 0.632$ cm, 1 m $= 0.0632$ cm).

1. Draw a line 9.49 cm long and divide it into main divisions of 50 m ($3.16$ cm each): three divisions for 150 m. Add one more division to the left of 0.
2. Divide the left main division into 5 sub-divisions of 10 m each ($0.632$ cm).
3. Draw 10 equally spaced horizontal lines on a rectangle about 2 cm high, and draw diagonals from the top of each sub-division to the next lower sub-division on the base. Each step on a diagonal reads 1 m (a diagonal scale is needed because 1 m is only 0.63 mm).

```text
 10 |  /  /  /  /  /  /  /  /  /  |
  5 | /  /                        |
  0 +---+-----+-----+-------------+
   50..0   50   100   150 (m)
```

#### Area on the 1:12500 sheet
Linear ratio $= 1/12500$, so the same ground area shown on the topo sheet is

$$
\text{Area on sheet} = \frac{18000 \times 10^{4}\ \text{cm}^2}{12500^{2}} = 1.152\ \text{cm}^2
$$

The plan would measure about $1.01\ \text{cm} \times 1.14\ \text{cm}$ on the sheet (each side of 8 cm and 9 cm is reduced by the factor $1581.1/12500$).

**Answer: RF $\approx$ 1/1581 (1 cm = 15.81 m); area on the 1:12500 sheet = 1.152 cm$^2$.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Map area 72 cm^2 and ground area 1.8 x 10^8 cm^2 give which scale?
   A. About 1:1581
   B. About 1:2500
   C. About 1:15810
   D. 1:12500
   Answer: A. RF = sqrt(72 / 1.8 x 10^8) = 6.32 x 10^-4, about 1/1581 (1 cm = 15.81 m). (Key point: RF from areas)

2. The same ground area is drawn at 1:12500. How is its area on the sheet found?
   A. Ground area / 12500
   B. Ground area / 12500^2 (in cm^2)
   C. Original map area x 12500
   D. Original map area / 12500
   Answer: B. Areas scale with the square of the linear ratio, so area = 1.8 x 10^8 / 12500^2 = 1.152 cm^2. (Key point: Area on a new scale)

3. For a maximum reading of 150 m at RF 1/1581, what is the scale length?
   A. 15 cm
   B. 94.9 cm
   C. 9.49 cm
   D. 0.949 cm
   Answer: C. 150 x 100 / 1581.1 = 9.49 cm. (Key point: Scale length)

4. Main divisions are 50 m. How is the extra left division split?
   A. Into 10 parts of 5 m
   B. Into 5 parts of 10 m
   C. Into 50 parts of 1 m
   D. Into 2 parts of 25 m
   Answer: B. The 50 m division is split into 5 sub-divisions of 10 m, which diagonals then divide into 1 m. (Key point: Sub-divisions)

## Related in Surveying I

- [Scale for 39672 m² plan](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Ascale-plot-39672-sqm) — Surveying I, chapter 1
- [RF and scale for 6.304 ha plot](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Ascale-plot-6-304-ha) — Surveying I, chapter 1, asked 3×
- [RF and scale for 51750 m² plot](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Ascale-plot-51750-sqm) — Surveying I, chapter 1, asked 3×
- [RF and scale for 51072 m² plot](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Ascale-plot-51072-sqm) — Surveying I, chapter 1, asked 3×
- [RF and scale for 0.55 hectare plot](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Ascale-plot-0-55-hectare) — Surveying I, chapter 1
- [Scale for 1.5 km road on map](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Ascale-road-1-5-km) — Surveying I, chapter 1

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