UTM Zone of a Control Point: 18-Geom-A1 Surveying, May 2019, Q7

Question 7: UTM Zone of a Control Point (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Latitude $\phi = 43^\circ10'30.205''$ N; longitude $\lambda = 121^\circ30'45.638''$ W (i.e. $\lambda=-121.5127^\circ$).

Find. The UTM zone number (and hemisphere designation) containing BM.1.

A 6°-wide UTM zone: scale factor 0.9996 on the central meridian, unity on the two standard meridians ~180 km either side.

Approach. UTM divides the globe into 60 zones each $6^\circ$ wide, numbered 1–60 eastward from the antimeridian ($180^\circ$W). Apply the zone formula to the longitude.

Longitude in decimal degrees. $$ \lambda = -\left(121 + \tfrac{30}{60} + \tfrac{45.638}{3600}\right) = -121.5127^\circ. $$
Zone number. $$ Z = \left\lfloor \frac{\lambda + 180^\circ}{6^\circ} \right\rfloor + 1 = \left\lfloor \frac{58.4873}{6} \right\rfloor + 1 = 9 + 1 \;\Rightarrow\; \boxed{Z = 10} $$
Extent and hemisphere. Zone 10 spans $126^\circ$W to $120^\circ$W with central meridian $123^\circ$W; $121.5^\circ$W lies within it, $1.5^\circ$ east of the CM. As $\phi$ is north, the designation is Zone 10N (MGRS latitude band T, $40^\circ$–$48^\circ$N).

Quantity	Value
Longitude (decimal)	$-121.5127^\circ$
UTM zone	$10$ (10N)
Central meridian	$123^\circ$W
MGRS latitude band	T

Concept & Theory

Topic. The Universal Transverse Mercator zone system.

Key relations. Zone $Z=\lfloor(\lambda+180^\circ)/6\rfloor+1$; central meridian $\lambda_0 = -183^\circ + 6Z$; scale factor $k_0=0.9996$ at the CM.

Why this works. Each zone is a $6^\circ$ strip counted from the $180^\circ$ antimeridian, so shifting longitude by $+180^\circ$ and dividing by 6 indexes the strip directly; adding 1 makes the count start at 1. A secant projection ($k_0=0.9996$) keeps scale distortion under $\sim$1 part in 2500 across the zone.

Common pitfall. Dropping the sign of a west longitude, or forgetting the “+1” so the zone comes out one low. In Canada the common zones run 7–22; Zone 10 also covers southern British Columbia, although this particular point, at $43^\circ$N, lies in central Oregon, south of the Canadian border.

Source. Wolf & Ghilani, Elementary Surveying, 15th ed., §20.8 (UTM); NRCan, NAD83(CSRS) UTM specifications.

Question 7 of 9: UTM Zone of a Control Point

Question 7: UTM Zone of a Control Point (10 marks)