---
title: "Road earthwork, chainage 0-80 (s)"
subject: "Estimating and Costing"
chapter: 6
key: "semester_7/estimating_and_costing:road-earthwork-s-table-0-80"
source: https://www.bctengineer.com/q/semester_7/estimating_and_costing/road-earthwork-s-table-0-80
site: BCT Engineering
---

# Road earthwork, chainage 0-80 (s)

Estimating and Costing · Chapter 6: Detailed Estimate

- Asked in: 2067 Asar (old course), 7 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/estimating_and_costing/road-earthwork-s-table-0-80
- Whole chapter: https://www.bctengineer.com/md/semester_7/estimating_and_costing/chapter-6

## Question

Estimate the quantity of earthwork in cutting and filling from the following data for a portion of road. Formation width = 10 m, side slope in banking = 2:1, side slope in cutting = 1:1.

| Chainage | Depth of cutting (m) | Height of filling (m) | Cross slope of ground |
|---|---|---|---|
| 0 | 0.60 | — | 10:1 |
| 20 | 0.30 | — | 8:1 |
| 40 | 0.50 | — | 12:1 |
| 60 | — | 0.35 | 10:1 |
| 80 | — | 0.70 | 12:1 |

## Short answer

Apply side-hill section formulas for given formation width 10 m, side slopes, and transverse slopes at 20 m intervals. Earthwork quantities are 256.73 m³ in cutting and 187.23 m³ in filling.

## Answer

#### Given data and assumptions
- Formation width $B = 10$ m; side slope in banking $2:1$, in cutting $1:1$.
- Sections 20 m apart; cutting up to 40 m and filling at 60 m and 80 m.

#### Method
Ground has a transverse (cross) slope $n:1$ (H:V), so each section is a side-hill section. With centre height $h$ (cutting +, filling −) above/below formation level, formation width $B$ and side slope $s:1$:

$$
\begin{aligned} x_u &= \frac{n\,(B/2 + s h)}{n - s},\quad d_u = h + \frac{x_u}{n}\\ x_d &= \frac{n\,(B/2 + s h)}{n + s},\quad d_d = h - \frac{x_d}{n}\\ A &= \tfrac12 B h + \tfrac12\left[\tfrac{B}{2}(d_u + d_d) + h\,(x_u + x_d - B)\right] \end{aligned}
$$

Here $x_u, x_d$ are the horizontal distances of the up-hill and down-hill toes from the centre line and $d_u, d_d$ the heights of the ground above formation at the toes (formulas for cutting). For a section wholly in filling the roles swap: the down-hill toe is the far one, $x_d = n(B/2 + s h)/(n - s)$ and $x_u = n(B/2 + s h)/(n + s)$, with $h$ the filling height and $s$ the filling slope. Where the ground line crosses the formation level inside the road width (part cutting, part filling), the cutting and filling areas are found separately from the co-ordinates of the section (toe points from the same intersection equations). Volume between sections = mean area × distance, taken separately for cutting and filling.

#### Sectional areas
| Chainage | Centre ht. $h$ (m) | Cutting area (m²) | Filling area (m²) |
|---|---:|---:|---:|
| 0+000 | cut 0.60 | 6.677 | 0.000 |
| 0+020 | cut 0.30 | 3.911 | 0.563 |
| 0+040 | cut 0.50 | 5.462 | 0.000 |
| 0+060 | fill 0.35 | 0.125 | 4.516 |
| 0+080 | fill 0.70 | 0.000 | 8.565 |

Working for one section:

Take chainage 0+020: centre cutting depth $h = 0.30$ m, $n = 8$, $B = 10.0$ m.
- Ground height above formation at the uphill edge $= 0.925$ m, at the downhill edge $= -0.325$ m.
- Toe of the side slope on the uphill side at $x = 6.057$ m from the centre line (ground above formation by 1.057 m); downhill toe at $x = 5.867$ m (ground below formation by 0.433 m).
- Area from the co-ordinates of the polygon: cutting $= 3.911$ m², filling $= 0.563$ m².

#### Volumes
| Between | Distance (m) | Cutting (m³) | Filling (m³) |
|---|---:|---:|---:|
| 0+000 – 0+020 | 20.0 | 105.88 | 5.63 |
| 0+020 – 0+040 | 20.0 | 93.73 | 5.63 |
| 0+040 – 0+060 | 20.0 | 55.87 | 45.16 |
| 0+060 – 0+080 | 20.0 | 1.25 | 130.81 |
| **Total** | 80.0 | **256.73** | **187.23** |

**Answer: Earthwork in cutting = 256.73 m³; earthwork in filling = 187.23 m³.**

## Check yourself

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

1. With chainages 0, 20, 40, 60, 80, what distance is used in the mean-area volume?
   A. 80 m
   B. 40 m
   C. 10 m
   D. 20 m
   Answer: D. Each pair of consecutive sections is 20 m apart. (Key point: Spacing)

2. At 0+020 (cut 0.30 m, cross slope 8:1, B = 10 m) the down-hill edge is at what height relative to formation?
   A. +0.325 m
   B. -0.325 m
   C. -0.925 m
   D. +0.925 m
   Answer: B. Down-hill edge = 0.30 - 5/8 = -0.325 m, so a small filling area appears. (Key point: Filling part in cutting section)

3. Cutting areas 6.677 and 3.911 m^2, 20 m apart, give a cutting volume of about:
   A. 105.9 m^3
   B. 52.9 m^3
   C. 211.8 m^3
   D. 78.2 m^3
   Answer: A. (6.677 + 3.911)/2 = 5.294 m^2 x 20 = about 105.88 m^3. (Key point: Mean-area volume)

## Close variants set in other exams

- [Road earthwork, 80 m length (v)](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Aroad-earthwork-v-0-80) — Estimating and Costing, chapter 6, asked 2×

## Related in Estimating and Costing

- [Hill road earthwork, cutting 4 m / banking 6 m](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-k-cut4-fill6) — Estimating and Costing, chapter 6
- [Hill road earthwork, cross slope 1 in 5 (0-60)](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-r-cross-slope-1-5) — Estimating and Costing, chapter 6
- [Hill road earthwork, chainage 130-280](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-formation-8-table-a) — Estimating and Costing, chapter 6
- [Hill road earthwork, chainage 0-150 (50 m)](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-j-formation-10) — Estimating and Costing, chapter 6
- [Hill road earthwork, chainage 12-15](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-q-chain-12-15) — Estimating and Costing, chapter 6
- [Hill road earthwork, 0-90 m (8 m formation)](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-p-0-90) — Estimating and Costing, chapter 6

## Related in other subjects

- [Cutting volume, 60 m, GL 540.70, formation 538.20](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Avolume-cutting-60m-gl-540-70) — Surveying I, chapter 9, asked 7×
- [Cutting volume, 60 m, GL 1181.50, formation 1179.00](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Avolume-cutting-60m-gl-1181-50-formation-1179) — Surveying I, chapter 9, asked 8×
- [Cutting volume, 60 m, formation level 279.00](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Avolume-cutting-60m-gl-281-50-formation-279) — Surveying I, chapter 9, asked 7×
- [Cutting volume, 60 m, formation level 278.00](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Avolume-cutting-60m-formation-278-upward) — Surveying I, chapter 9, asked 7×