---
title: "Hill road earthwork, 0 m to 90 m, 10/8 m widths"
subject: "Estimating and Costing"
chapter: 6
key: "semester_7/estimating_and_costing:hill-road-e-0-90"
source: https://www.bctengineer.com/q/semester_7/estimating_and_costing/hill-road-e-0-90
site: BCT Engineering
---

# Hill road earthwork, 0 m to 90 m, 10/8 m widths

Estimating and Costing · Chapter 6: Detailed Estimate

- Asked in: 2080 Bhadra, 8 marks
- Video: [Calculation of Volume of Earthwork from Cross Sections | Road Estimate | Earthwork Calculation](https://www.youtube.com/watch?v=zMl6OVLeKWM) — Civil Engineering Edu
- 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-e-0-90
- Whole chapter: https://www.bctengineer.com/md/semester_7/estimating_and_costing/chapter-6

## Question

Calculate the volume of earthwork of a portion of hill road from the given data: Top formation width in cutting = 8 m, top formation width in banking = 10 m, side slope in cutting and banking are 1.5:1 and 2:1 (H:V), respectively.

| Distance | Height of Filling | Depth of Cutting | Transverse Slope of Ground (H:V) |
|---|---|---|---|
| 0 m | - | 0.60 | 10:1 |
| 30 m | - | 0.70 | 15:1 |
| 60 m | 0.80 | - | 12:1 |
| 90 m | 0.85 | - | 10:1 |

## Short answer

Compute sectional areas using side-hill formulae. Prorate the cut-fill transition length by centre heights. Mean area volumes: cutting 228.78 m³, filling 390.96 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 30 m apart. Cutting at 0 m and 30 m, filling at 60 m and 90 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+000 | cut 0.60 | 5.708 | 0.000 |
| 0+030 | cut 0.70 | 6.507 | 0.000 |
| 0+060 | fill 0.80 | 0.000 | 9.902 |
| 0+090 | fill 0.85 | 0.000 | 10.880 |

Working for one section:

Take chainage 0+090: centre filling height $h = 0.85$ m, $n = 10$, $B = 10.0$ m.
- Ground height above formation at the uphill edge $= -0.350$ m, at the downhill edge $= -1.350$ m.
- Toe of the side slope on the uphill side at $x = 5.583$ m from the centre line (ground below formation by 0.292 m); downhill toe at $x = 8.375$ m (ground below formation by 1.688 m).
- Area from the co-ordinates of the polygon: cutting $= 0.000$ m², filling $= 10.880$ m².

Change-over between 0+030 (wholly in cutting, 0.70 m) and 0+060 (wholly in filling, 0.80 m): the zero point is $30 \times 0.70/1.50 = 14.00$ m from 0+030, leaving $16.00$ m in filling.
- Cutting $= \tfrac12 \times 6.507 \times 14.00 = 45.55$ m³
- Filling $= \tfrac12 \times 9.902 \times 16.00 = 79.22$ m³

#### Volumes
| Between | Distance (m) | Cutting (m³) | Filling (m³) |
|---|---:|---:|---:|
| 0+000 – 0+030 | 30.0 | 183.23 | 0.00 |
| 0+030 – 0+060 | 30.0 (14.00 cut + 16.00 fill) | 45.55 | 79.22 |
| 0+060 – 0+090 | 30.0 | 0.00 | 311.74 |
| **Total** | 90.0 | **228.78** | **390.96** |

**Answer: Earthwork in cutting = 228.78 m³; earthwork in filling = 390.96 m³.**

## Check yourself

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

1. Which sections of this hill road are in cutting and which in filling?
   A. 0 and 30 m in cutting; 60 and 90 m in filling
   B. 0 m in cutting; 30, 60 and 90 m in filling
   C. All in cutting
   D. 0, 30 and 60 m in cutting; 90 m in filling
   Answer: A. Depth of cutting is given at 0 and 30 m and height of filling at 60 and 90 m. (Key point: Section types)

2. What formation width B is used for the section at 0+090?
   A. 8 m
   B. 10 m
   C. 18 m
   D. 9 m
   Answer: B. Filling (banking) sections use the banking formation width of 10 m. (Key point: Formation width per section)

3. For a section wholly in filling, the far (down-hill) toe distance is:
   A. n(B/2 + s h)/(n + s)
   B. n(B/2 + s h)/(n - s)
   C. (B/2 + s h)/n
   D. B/2 + n h
   Answer: B. In filling the roles swap: the downhill toe is the far one and uses (n - s); the uphill toe uses (n + s). (Key point: Filling toe formula)

4. In the volume table, the transition 0+030 to 0+060 shows 97.60 m3 cutting and 148.53 m3 filling. Why are both present?
   A. The row mixes two chainages' areas: cutting area from the cut end and filling area from the fill end, each averaged with zero at the other end
   B. It is an error
   C. Cutting is carried over as filling
   D. The ground slope changes
   Answer: A. Cutting area at 0+030 is averaged with 0 at 0+060, and filling area at 0+060 with 0 at 0+030, each times 30 m (mean area x distance, separately). (Key point: Totals)

## Close variants set in other exams

- [Hill road earthwork, 0+020 to 0+140](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-c-three-sections) — Estimating and Costing, chapter 6, asked 2×

## Related in Estimating and Costing

- [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, 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
- [Side-long ground hill road, 0-400 m](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Aside-long-road-n) — Estimating and Costing, chapter 6
- [Hill road earthwork, 0+060 to 0+120](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-m-formation-10-8) — 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, 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

## 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
- [Cut and fill volume, 20 m formation width](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Avolume-road-20m-width-cut-fill-1-1-1-2) — 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×
- [Filling volume, 50 m, formation 1081.50](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Avolume-filling-50m-gl-1080-50) — Surveying I, chapter 9, asked 4×