---
title: "Cutting volume, 120 m, upward slope 1 in 40"
subject: "Surveying I"
chapter: 9
key: "semester_3/surveying_i:volume-cutting-120m-formation-380"
source: https://www.bctengineer.com/q/semester_3/surveying_i/volume-cutting-120m-formation-380
site: BCT Engineering
---

# Cutting volume, 120 m, upward slope 1 in 40

Surveying I · Chapter 9: Computation of Area and Volume

- Asked in: 2081 Baisakh, 6 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/surveying_i/volume-cutting-120m-formation-380
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-9

## Question

Find the volume in a length of 120 m with the following data for a two level section using the prismoidal and trapezoidal formula. Also calculate the prismoidal correction. Formation width = 8 m, side slope is 1.5:1, transverse slope = 6:1. The ground level at 30 m intervals are given below.

| Chainage (m) | 0 | 30 | 60 | 90 | 120 |
|---|---|---|---|---|---|
| Ground level (m) | 380.8 | 382.3 | 383.6 | 384.1 | 385.6 |

The formation has an upward slope of 1 in 40 with the formation level at 0+000 chainage being 380.00 m.

## Short answer

With two-level sections, centre depths 0.80–2.60 m, areas 8.562–33.714 m², spacing 30 m. Trapezoidal volume = 2597.43 m³, prismoidal = 2526.74 m³, prismoidal correction = 70.69 m³.

## Answer

A two-level section (ground sloping across the road) is used at every chainage. Data: formation width $b = 8$ m, side slope $s:1 = 1.5:1$ (horizontal : vertical), transverse ground slope $n:1 = 6:1$, section spacing $L = 30$ m.

#### Step 1: Formation levels and depths at the centre line

The formation rises at 1 in 40, so it gains $30/40 = 0.75$ m in every 30 m: $FL = 380.00 + 0.75 \times (\text{number of intervals})$. The centre depth of cut is $h = GL - FL$.

| Chainage (m) | GL (m) | FL (m) | Centre depth of cut h (m) |
|---|---|---|---|
| 0 | 380.80 | 380.00 | 0.80 |
| 30 | 382.30 | 380.75 | 1.55 |
| 60 | 383.60 | 381.50 | 2.10 |
| 90 | 384.10 | 382.25 | 1.85 |
| 120 | 385.60 | 383.00 | 2.60 |

#### Step 2: Cross-sectional areas

Along the cross-section the ground rises on one side and falls on the other. The depths of cut at the two side slope ends (where the side slope meets the ground) are

$$
h_1 = \frac{nh + b/2}{n - s} \quad (\text{high-ground side}), \qquad h_2 = \frac{nh - b/2}{n + s} \quad (\text{low-ground side})
$$

The horizontal distances from the centre line to these points are $d_1 = b/2 + s h_1$ and $d_2 = b/2 + s h_2$, and the area of the section is

$$
A = \frac{1}{2}(d_1 + d_2)\left(h + \frac{b}{2s}\right) - \frac{b^2}{4s}
$$

(the triangle between the two side slopes extended and the ground line, minus the triangle below the formation width).

Worked section at chainage 0, $h = 0.80$ m:

$$
\begin{aligned} h_1 &= \frac{6 \times 0.80 + 4}{6 - 1.5} = 1.956\ \text{m}, & h_2 &= \frac{6 \times 0.80 - 4}{6 + 1.5} = 0.107\ \text{m} \\ d_1 &= 4 + 1.5 \times 1.956 = 6.933\ \text{m}, & d_2 &= 4 + 1.5 \times 0.107 = 4.160\ \text{m} \\ A &= \tfrac12(6.933 + 4.160)\left(0.80 + \tfrac{8}{3}\right) - \tfrac{64}{6} = 8.562\ \text{m}^2 \end{aligned}
$$

| Chainage (m) | h (m) | h₁ (m) | h₂ (m) | d₁ (m) | d₂ (m) | Area A (m²) |
|---|---|---|---|---|---|---|
| 0 | 0.80 | 1.956 | 0.107 | 6.933 | 4.160 | 8.562 |
| 30 | 1.55 | 2.956 | 0.707 | 8.433 | 5.060 | 17.782 |
| 60 | 2.10 | 3.689 | 1.147 | 9.533 | 5.720 | 25.687 |
| 90 | 1.85 | 3.356 | 0.947 | 9.033 | 5.420 | 21.974 |
| 120 | 2.60 | 4.356 | 1.547 | 10.533 | 6.320 | 33.714 |

#### Step 3: Volume by the trapezoidal formula

$$
V_T = L\left[\frac{A_1 + A_{5}}{2} + A_2 + \dots + A_{4}\right] = 30\left[\frac{8.562 + 33.714}{2} + 17.782 + 25.687 + 21.974\right] = 2597.43\ \text{m}^3
$$

#### Step 4: Volume by the prismoidal formula

$$
V_P = \frac{L}{3}\left[A_1 + A_{5} + 4(A_2 + A_4) + 2A_3\right] = \frac{30}{3}\left[8.562 + 33.714 + 4(17.782 + 21.974) + 2(25.687)\right] = 2526.74\ \text{m}^3
$$

#### Step 5: Prismoidal correction

$$
C_p = V_T - V_P = 2597.43 - 2526.74 = 70.69\ \text{m}^3
$$

The trapezoidal formula overestimates the volume by 70.69 m³, so the prismoidal volume is the more accurate one.

**Answer: Volume of cutting by the trapezoidal formula = 2597.43 m³; by the prismoidal formula = 2526.74 m³; prismoidal correction = 70.69 m³.**

## Check yourself

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

1. The formation rises 1 in 40 starting at 380.00 m. What is FL at 30 m?
   A. 380.30 m
   B. 380.75 m
   C. 381.50 m
   D. 380.00 m
   Answer: B. Rise = 30/40 = 0.75 m per 30 m. (Key point: Formation level)

2. How is the centre depth of cut obtained?
   A. h = ground level - formation level
   B. h = formation level - ground level
   C. h = average of both ground levels
   D. h = transverse slope x width
   Answer: A. For a cutting, ground is above formation, so h = GL - FL. (Key point: Centre depth)

3. Which formula gives the prismoidal volume for 5 sections?
   A. L/2 [A1 + 2A2 + A3]
   B. L [A1 + A5 + A2 + A3 + A4]
   C. L/3 [A1 + A5 + 2(A2 + A4) + 4A3]
   D. L/3 [A1 + A5 + 4(A2 + A4) + 2A3]
   Answer: D. Simpson-type rule: odd-numbered sections (2nd, 4th) multiply by 4 and middle one by 2. (Key point: Prismoidal formula)

4. What is prismoidal correction here and what does it mean?
   A. Cp = V_prismoidal - V_trapezoidal; prismoidal overestimates
   B. Cp = 0 since sections are equal
   C. Cp = V_trapezoidal - V_prismoidal = 70.69 m^3; trapezoidal overestimates
   D. Cp = 70.69 m^3 and trapezoidal underestimates
   Answer: C. 2597.43 - 2526.74 = 70.69 m^3; the trapezoidal volume is too large. (Key point: Prismoidal correction)

## Close variants set in other exams

- [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 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×
- [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, 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×

## Related in Surveying I

- [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
- [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
- [Filling volume, 60 m, formation 1281.49](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Avolume-filling-60m-gl-1280-50) — Surveying I, chapter 9
- [Cutting volume, depths 2 m, 3 m, 4 m](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Avolume-cutting-8m-depths-2-3-4) — Surveying I, chapter 9
- [Prismoidal formula for volume](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aprismoidal-formula-volume) — Surveying I, chapter 9
- [Volume from central depths 1.85, 2.15, 2.45](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Avolume-central-depths-1-85-2-15-2-45) — Surveying I, chapter 9

## Related in other subjects

- [Road earthwork, formation level at 150 m](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Aroad-earthwork-t-grade-150) — Estimating and Costing, chapter 6, asked 2×
- [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
- [Economical trapezoidal channel side length](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Aeconomical-trapezoidal-side-length) — Hydro Power, chapter 1
- [Vertical drop crest level and width, Q 30](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Avertical-drop-crest-q30-fsl315) — Irrigation and Drainage Engineering, chapter 7