---
title: "Closed traverse missing EA bearing, BC length"
subject: "Surveying II"
chapter: 1
key: "semester_4/surveying_ii:closed-traverse-abcdea-missing-ea-bc"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/closed-traverse-abcdea-missing-ea-bc
site: BCT Engineering
---

# Closed traverse missing EA bearing, BC length

Surveying II · Chapter 1: Traversing

- Asked in: 2080 Chaitra, 3+3+2+2 marks
- Video: [omitted measurement of closed traverse || omitted measurement|| missing length & bearing in traverse](https://www.youtube.com/watch?v=iBV3daNWtaY) — Aim 2 Civil Class
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/closed-traverse-abcdea-missing-ea-bc
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-1

## Question

The bearings and lengths of the sides of a closed traverse ABCDEA are given below but bearing of EA and length of BC are missed, find those missing figures. Also compute the coordinates of other points if coordinates of C are: Northing of C = 1000.00 m, Easting of C = 1000.00 m.

| Line | AB | BC | CD | DE | EA |
|---|---|---|---|---|---|
| Length (m) | 282.20 | ? | 324.70 | 381.60 | 359.60 |
| Bearings | 61°30' | 151°24' | 201°02' | 280°14' | ? |

## Short answer

Set closure equations ΣLat=0, ΣDep=0 for the known lines; solve quadratic for BC length: 274.321 m. Then compute lat/dep of EA, bearing = arctan(112.751/341.467) = 18°16' (NE quadrant). BC = 274.321 m, EA bearing = 18°16'. With C (1000 N, 1000 E), coordinates: A (1106.195 N, 620.683 E), B (1240.849 N, 868.685 E), D (696.934 N, 883.462 E), E (764.728 N, 507.932 E).

## Answer

For a closed traverse $\Sigma\text{Lat}=0$ and $\Sigma\text{Dep}=0$. Two quantities are unknown, the length of BC and the bearing of EA, so the two closure equations are enough to find them.

#### Step 1: Latitudes and departures of the known lines
$$
\text{Lat}=L\cos\theta,\qquad \text{Dep}=L\sin\theta
$$

| Line | Length (m) | Bearing | Latitude | Departure |
|---|---|---|---|---|
| AB | 282.200 | 61°30' | +134.654 | +248.002 |
| CD | 324.700 | 201°02' | -303.066 | -116.538 |
| DE | 381.600 | 280°14' | +67.794 | -375.530 |
| **Sum** |  |  | -100.618 | -244.066 |

Let the unknown length of BC be $x$ (bearing 151°24'). Its latitude is $x\cos\theta$ and its departure $x\sin\theta$. The line EA has length 359.60 m and must close the traverse, so

$$
((-100.618)+x\cos\theta)^2+((-244.066)+x\sin\theta)^2=359.60^2
$$

with $\cos\theta=-0.87798$ and $\sin\theta=0.47869$ for 151°24'.

#### Step 2: Solve for the length
$$
x^2+2((-100.618)\times(-0.87798)+(-244.066)\times0.47869)x+((-100.618)^2+(-244.066)^2-359.60^2)=0
$$

$$
x^2+(-56.984)x+(-59620.10)=0
$$

$$
x=\frac{-(-56.984)\pm\sqrt{241727.5}}{2}=274.321\ \text{or}\ -217.337
$$

The negative root has no meaning, so **length of BC = 274.321 m**.

#### Step 3: Bearing of EA
Latitude of EA $=-((-100.618)+274.321\times(-0.87798))=341.467$ m and departure $=-((-244.066)+274.321\times0.47869)=112.751$ m.

$$
\tan\theta=\frac{112.751}{341.467}\ \Rightarrow\ \theta=18^\circ 16'
$$

Both the latitude and departure are positive, so the line lies in the north-east quadrant and the whole circle bearing is **18°16'** (N 18°16' E).

#### Step 4: Check
| Line | Length (m) | Bearing | Latitude | Departure |
|---|---|---|---|---|
| AB | 282.200 | 61°30' | +134.654 | +248.002 |
| BC | 274.321 | 151°24' | -240.849 | +131.315 |
| CD | 324.700 | 201°02' | -303.066 | -116.538 |
| DE | 381.600 | 280°14' | +67.794 | -375.530 |
| EA | 359.600 | 18°16' | +341.467 | +112.751 |
| **Sum** |  |  | +0.000 | +0.000 |

Both sums are zero, so the traverse closes.

#### Coordinates of the other stations (C = 1000.000 N, 1000.000 E)
Independent coordinates = previous coordinates + latitude/departure of the line, going C, D, E, A, B, C:

| Station | Northing (m) | Easting (m) |
|---|---|---|
| A | 1106.195 | 620.683 |
| B | 1240.849 | 868.685 |
| C | 1000.000 | 1000.000 |
| D | 696.934 | 883.462 |
| E | 764.728 | 507.932 |

Check: B plus line BC returns exactly to C (1000.000 N, 1000.000 E).

**Answer:** length of BC = 274.321 m; bearing of EA = 18°16'.

## Check yourself

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

1. Length of BC and bearing of EA are unknown. Why can both be found?
   A. Two closure conditions (sum Lat = 0, sum Dep = 0) give two equations for two unknowns
   B. Angles alone give them
   C. Bowditch's rule gives them
   D. BC equals EA
   Answer: A. Closure of latitudes and departures supplies two equations, enough for two unknowns. (Key point: Unknowns and equations)

2. The quadratic gives x = 274.321 or -217.337. Which is taken for BC and why?
   A. -217.337, as it is smaller in magnitude
   B. Their mean
   C. 274.321, because a negative length is meaningless
   D. Neither, as the data are inconsistent
   Answer: C. A length cannot be negative, so BC = 274.321 m. (Key point: Quadratic solution)

3. EA has latitude +341.467 and departure +112.751. Its bearing is:
   A. 71°44'
   B. 161°44'
   C. 341°44'
   D. 18°16'
   Answer: D. Both components are positive (NE quadrant): tan^-1(112.751/341.467) = 18°16'. (Key point: Bearing of EA)

4. How is the final solution verified?
   A. Latitudes and departures of all five lines each sum to zero
   B. All lines have equal length
   C. Bearings add to 360°
   D. Sum of lengths equals 1000
   Answer: A. The check table shows sum Lat = 0 and sum Dep = 0, so the traverse closes. (Key point: Closure check)

## Related in Surveying II

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- [Traverse missing length CD, bearing EA](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atraverse-missing-cd-length-ea-bearing) — Surveying II, chapter 1
- [Missing bearings of BC and DE in traverse](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atraverse-missing-bearings-bc-de) — Surveying II, chapter 1
- [Bearings of DE and EA in traverse ABCDE](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atraverse-abcde-bearings-de-ea) — Surveying II, chapter 1, asked 2×
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