---
title: "Hill road earthwork, 0+020 to 0+140"
subject: "Estimating and Costing"
chapter: 6
key: "semester_7/estimating_and_costing:hill-road-c-three-sections"
source: https://www.bctengineer.com/q/semester_7/estimating_and_costing/hill-road-c-three-sections
site: BCT Engineering
---

# Hill road earthwork, 0+020 to 0+140

Estimating and Costing · Chapter 6: Detailed Estimate

- Asked in: 2081 Bhadra, 8 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/hill-road-c-three-sections
- Whole chapter: https://www.bctengineer.com/md/semester_7/estimating_and_costing/chapter-6

## Question

Data for a hill road portion is given below. Calculate the volume of earthwork.

| Chainage | Depth of Cutting (m) | Height of banking (m) | Transverse Slope of Ground |
|---|---|---|---|
| 0+020 m | 0.6 | - | 10:1 |
| 0+070 m | 0.7 | - | 15:1 |
| 0+140 m | - | 0.8 | 12:1 |

Width in cutting = 8 m, Formation width in banking = 10 m, Side slope in cutting = 1.5:1 (H:V), Side slope in banking = 2:1 (H:V).

## Short answer

Use side-hill section formulas to find areas. Where cutting changes to filling, prorate length by centre heights. Volume by mean area method gives cutting 411.66 m³, filling 184.84 m³.

## Answer

#### Given data and assumptions
- Formation width: cutting $8$ m, banking $10$ m; side slopes: cutting $1.5:1$, banking $2:1$ (H:V).
- Sections at 0+020, 0+070 and 0+140 m, so the distances are 50 m and 70 m.

#### Method
The 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 $d_u, d_d$ are 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. Where the centre changes from cutting to filling between two sections and one type of area is zero at one end, the change-over point is found by proportion of the centre heights, and that area is taken over its own part of the length only: $V = \tfrac12 A \times l$.

#### Sectional areas
| Chainage | Centre ht. $h$ (m) | Cutting area (m²) | Filling area (m²) |
|---|---:|---:|---:|
| 0+020 | cut 0.60 | 5.708 | 0.000 |
| 0+070 | cut 0.70 | 6.507 | 0.000 |
| 0+140 | fill 0.80 | 0.000 | 9.902 |

Working for one section:

Take chainage 0+020: centre cutting depth $h = 0.60$ m, $n = 10$, $B = 8.0$ m.
- Ground height above formation at the uphill edge $= 1.000$ m, at the downhill edge $= 0.200$ m.
- Toe of the side slope on the uphill side at $x = 5.765$ m from the centre line (ground above formation by 1.176 m); downhill toe at $x = 4.261$ m (ground above formation by 0.174 m).
- Area from the co-ordinates of the polygon: cutting $= 5.708$ m², filling $= 0.000$ m².

Change-over between 0+070 (wholly in cutting, 0.70 m) and 0+140 (wholly in filling, 0.80 m): the zero point is $70 \times 0.70/1.50 = 32.67$ m from 0+070, leaving $37.33$ m in filling.
- Cutting $= \tfrac12 \times 6.507 \times 32.67 = 106.28$ m³
- Filling $= \tfrac12 \times 9.902 \times 37.33 = 184.84$ m³

#### Volumes
| Between | Distance (m) | Cutting (m³) | Filling (m³) |
|---|---:|---:|---:|
| 0+020 – 0+070 | 50.0 | 305.38 | 0.00 |
| 0+070 – 0+140 | 70.0 (32.67 cut + 37.33 fill) | 106.28 | 184.84 |
| **Total** | 120.0 | **411.66** | **184.84** |

**Answer: Earthwork in cutting = 411.66 m³; earthwork in filling = 184.84 m³.**

## Check yourself

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

1. In this problem which formation width is used for the cutting sections and which for the banking section?
   A. 8 m for cutting and 10 m for banking
   B. 10 m for cutting and 8 m for banking
   C. 8 m for both
   D. 10 m for both
   Answer: A. The question states width in cutting is 8 m and in banking is 10 m, so B changes with the section type. (Key point: Different formation widths)

2. The sections are at 0+020, 0+070 and 0+140. The distances used for the mean-area volumes are:
   A. 30 m and 30 m
   B. 50 m and 70 m
   C. 20 m and 70 m
   D. 70 m and 140 m
   Answer: B. 0+070 - 0+020 = 50 m and 0+140 - 0+070 = 70 m. (Key point: Distance between sections)

3. How are the toe distances x_u and x_d on the uphill and downhill sides related in a cutting section with cross slope n and cut slope s?
   A. x_u = n(B/2 + s h)/(n + s) and x_d = n(B/2 + s h)/(n - s)
   B. x_u = n(B/2 + s h)/(n - s) and x_d = n(B/2 + s h)/(n + s)
   C. Both equal B/2 + s h
   D. Both equal n(B/2 + s h)
   Answer: B. Uphill the ground rises towards the cut face so the toe is farther, with (n - s) in the denominator; downhill it is nearer, with (n + s). (Key point: Side-hill area method)

4. The volume from 0+070 (cutting only) to 0+140 (filling only) is shown as 227.74 m3 cutting and 346.58 m3 filling. This means:
   A. Cutting and filling are both calculated for the 70 m length and the answer is the sum of the two
   B. Each type is computed separately for the transition, so both appear in that row
   C. Only the larger of the two is kept
   D. Filling is subtracted from cutting
   Answer: B. Cutting and filling volumes are taken separately between sections, even where the road changes from cut to fill. (Key point: Volume with a cut-to-fill change)

## Close variants set in other exams

- [Hill road earthwork, 0 m to 90 m, 10/8 m widths](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-e-0-90) — Estimating and Costing, chapter 6, asked 2×

## Related in Estimating and Costing

- [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, 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, 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, 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
- [Hill road earthwork, chainage 0-600 (12 marks)](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-l-0-600) — Estimating and Costing, chapter 6

## Related in other subjects

- [Earthwork volume, formation 1201.85, 1.65 m depth](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Avolume-earthwork-formation-1201-85) — Surveying I, chapter 9
- [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×