---
title: "Depth at fourth borehole P (coal seam)"
subject: "Engineering Geology II"
chapter: 6
key: "semester_4/engineering_geology_ii:bh-depth-at-p-coal-seam"
source: https://www.bctengineer.com/q/semester_4/engineering_geology_ii/bh-depth-at-p-coal-seam
site: BCT Engineering
---

# Depth at fourth borehole P (coal seam)

Engineering Geology II · Chapter 6: Measurement, analysis and interpretation of structural geological data

- Asked in: 2079 Asoj, 4+2 marks
- Video: [The three point problem](https://www.youtube.com/watch?v=d3wBJZZqQZw) — Rob Butler
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/engineering_geology_ii/bh-depth-at-p-coal-seam
- Whole chapter: https://www.bctengineer.com/md/semester_4/engineering_geology_ii/chapter-6

## Question

Fourth borehole is proposed at P, the NE corner of the square land. Calculate at what depth the borehole encounters the coal seam at P. [Data of the other three boreholes is not printed in this paper.]

## Short answer

For a plane seam in square borehole layout, fourth-corner depth equals sum of adjacent depths minus opposite: dP = dB + dC - dA. With A=15 m, B=45 m, C=60 m, P (NE) encounters the coal seam at 90 m.

## Answer

**Principle:** for a plane bed and a square (or any parallelogram) layout of vertical boreholes, the depth at the fourth corner is found from the other three corners:
$$
d_P = d_{\text{adjacent 1}} + d_{\text{adjacent 2}} - d_{\text{opposite}}
$$
This works because the depth changes uniformly along each side, so the depth at P equals the depth at the opposite corner plus the changes in depth along the two sides that meet there.

The depths of the other three boreholes are not printed with this part of the question. This is the follow-up to the companion problem with the same square layout (side 300 m), whose data are used here: SW corner A = 15 m, SE corner B = 45 m, NW corner C = 60 m, with P at the NE corner.

#### Calculation
- Gradient along the south side (A to B): $\dfrac{45 - 15}{300} = 0.10$ m/m
- Gradient along the west side (A to C): $\dfrac{60 - 15}{300} = 0.15$ m/m
- Depth at P (NE corner): $d_P = 15 + 0.10 \times 300 + 0.15 \times 300 = 90$ m

Check: $d_P = d_B + d_C - d_A = 45 + 60 - 15 = 90$ m.

#### Attitude (for completeness)
$\tan\delta = \sqrt{0.10^2 + 0.15^2} = 0.1803 \Rightarrow \delta = 10.2^\circ$; dip direction $= \arctan(0.10/0.15) = 33.7^\circ$ (NE); strike $303.7^\circ$ (N56.3°W).

**Answer: with the above data, the coal seam is met at P at a depth of 90 m. For any other data, use $d_P = d_B + d_C - d_A$ (the sum of the two adjacent corners minus the opposite corner).**

## Check yourself

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

1. Which formula gives the depth at the fourth corner?
   A. dP = dB + dC − dA, the sum of the two adjacent minus the opposite
   B. dP = dA + dB + dC
   C. dP = dB − dC
   D. dP = dA
   Answer: A. For a plane bed, depth varies linearly so opposite corners' sums are equal. (Key point: Parallelogram rule)

2. What gradients are used with A = 15, B = 45, C = 60 over 300 m?
   A. 0.15 and 0.10
   B. 0.10 and 0.15
   C. 0.3 and 0.45
   D. 0.05 and 0.20
   Answer: B. (45 − 15)/300 = 0.10 and (60 − 15)/300 = 0.15. (Key point: Gradients)

## Related in Engineering Geology II

- [Borehole attitude: square 360 m, M N O P](https://www.bctengineer.com/question.md?key=semester_4%2Fengineering_geology_ii%3Abh-attitude-square-360-mnop) — Engineering Geology II, chapter 6
- [Coal seam attitude: square 300 m](https://www.bctengineer.com/question.md?key=semester_4%2Fengineering_geology_ii%3Abh-coal-seam-square-300) — Engineering Geology II, chapter 6
- [Tunnel alignment from three boreholes](https://www.bctengineer.com/question.md?key=semester_4%2Fengineering_geology_ii%3Atunnel-alignment-three-boreholes) — Engineering Geology II, chapter 6
- [Limestone attitude: square 900 m, depth at D](https://www.bctengineer.com/question.md?key=semester_4%2Fengineering_geology_ii%3Abh-limestone-square-900-abcd) — Engineering Geology II, chapter 6

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