ECZ GRADE 12 & GCE MATHEMATICS

Earth Geometry: Latitudes, Longitudes & Nautical Distances

Master Section B (Question 11/12): calculating great and small circle distances, converting degrees to nautical miles, calculating airplane speeds in knots, and time zone offsets.

15 Min Read Paper 2 Section B (8 Marks) 100% Guaranteed Topic
Exam Candidate Blueprint

In ECZ Mathematics Paper 2, the final question in Section B (Question 12) is virtually always an 8-mark question on Earth Geometry. Because the question format and formulas are identical every single year, preparing this topic guarantees an automatic 8 marks out of 48 in Section B!

Teaching Guidance

Emphasize the difference between distances along a meridian (where the radius is the full Earth radius $R$) and distances along a parallel of latitude (where the radius is the smaller radius $r = R \cos \theta$). Drilling angle differences between same vs opposite hemispheres eliminates 90% of candidate errors.

Table of Contents

1. The Coordinate System: Latitudes vs Longitudes

Any location on Earth's spherical surface is specified by a pair of coordinates: (Latitude, Longitude), written in the order $(x^\circ\text{N/S}, y^\circ\text{E/W})$.

Parallels of Latitude
  • Horizontal circles running parallel to the Equator ($0^\circ$).
  • Measured from $0^\circ$ at the Equator up to $90^\circ\text{N}$ (North Pole) and down to $90^\circ\text{S}$ (South Pole).
  • Except for the Equator, all lines of latitude are small circles.
Meridians of Longitude
  • Vertical semi-circles running from the North Pole to the South Pole.
  • Measured east or west from the Greenwich Meridian ($0^\circ$) up to $180^\circ$.
  • Every meridian paired with its opposite forms a great circle.

2. How to Calculate the Angular Difference ($\theta$)

Before calculating distances, you must find the angle subtended at the center of the circle:

The Golden Hemisphere Rule:
  • Opposite Hemispheres (N and S, or E and W): ADD the angles.
    Example: Between $30^\circ\text{N}$ and $40^\circ\text{S}$, angular difference $\theta = 30^\circ + 40^\circ = 70^\circ$.
  • Same Hemisphere (N and N, S and S, or E and E): SUBTRACT the angles (larger minus smaller).
    Example: Between $55^\circ\text{E}$ and $20^\circ\text{E}$, angular difference $\theta = 55^\circ - 20^\circ = 35^\circ$.

3. Great Circles vs Small Circles

Understanding which circle you are moving along determines your radius:

  • Great Circle: Any circle whose center is the center of the Earth. Radius is Earth's radius $R$ ($R = 6370\text{ km}$ or $R = 3437\text{ nm}$). Movement along a meridian of longitude or along the Equator is along a great circle.
  • Small Circle: Any parallel of latitude $\theta^\circ\text{N}$ or $\theta^\circ\text{S}$ (other than the Equator). Its radius $r$ shrinks as you move toward the poles according to:
    $$r = R \cos \theta$$

4. Master Distance Formulas

Route / Circle Type Distance in Kilometres (km) Distance in Nautical Miles (nm)
Along a Meridian (Great Circle)
Two points with same longitude, different latitudes		 $$D = \frac{\theta}{360} \times 2\pi R$$ $$D = 60 \times \theta\text{ nm}$$
where 1° = 60 nm
Along a Parallel of Latitude (Small Circle)
Two points with same latitude $\alpha$, different longitudes		 $$D = \frac{\theta}{360} \times 2\pi R \cos \alpha$$ $$D = 60 \times \theta \cos \alpha\text{ nm}$$

In ECZ examinations, use $\pi = 3.142$ or $\frac{22}{7}$ and $R = 6370\text{ km}$ or $3437\text{ nm}$ as given on the examination paper cover.

5. Speed in Knots & Longitude Time Difference

Speed in Knots

In aviation and maritime navigation, 1 Knot is defined as 1 nautical mile per hour:

$$\text{Speed (knots)} = \frac{\text{Distance (nautical miles)}}{\text{Time (hours)}}$$
Time and Longitude

Earth rotates $360^\circ$ on its axis in 24 hours:

  • $15^\circ = 1\text{ hour (60 minutes)}$
  • $1^\circ = 4\text{ minutes}$
  • Places to the East are ahead in time (+).
  • Places to the West are behind in time (-).

6. Step-by-Step Worked Past Examination Question

ECZ Mathematics Paper 2 (8 Marks)

Given three points on Earth's surface: $P(60^\circ\text{N}, 40^\circ\text{E})$, $Q(60^\circ\text{N}, 20^\circ\text{W})$, and $R(30^\circ\text{S}, 40^\circ\text{E})$. [Take $\pi = 3.142$, $R = 6370\text{ km}$, and $1\text{ nautical mile} = 1.852\text{ km}$ or use $R = 3437\text{ nm}$]

Calculate:

  1. The difference in latitude between points $P$ and $R$. [1 Mark]
  2. The distance $PR$ along the common meridian in kilometres. [2 Marks]
  3. The distance $PQ$ along the latitude $60^\circ\text{N}$ in nautical miles. [3 Marks]
  4. An aeroplane flies from $P$ to $Q$ along the parallel of latitude in 5 hours. Calculate its speed in knots. [2 Marks]
Full Step-by-Step Solution:

Part (a): Difference in latitude between $P(60^\circ\text{N})$ and $R(30^\circ\text{S})$

Since they are in opposite hemispheres (North and South), add the latitudes: $$\theta = 60^\circ + 30^\circ = 90^\circ$$ Final Answer: $90^\circ$ [1 Mark]

Part (b): Distance $PR$ along the meridian in kilometres

Movement along a meridian is along a great circle: $$D = \frac{\theta}{360} \times 2\pi R$$ $$D = \frac{90}{360} \times 2 \times 3.142 \times 6370$$ $$D = \frac{1}{4} \times 40029.08 = 10,007.27\text{ km}$$ Final Answer: $10,000\text{ km}$ (or $10,010\text{ km}$ to 3 sig. figs) [2 Marks]

Part (c): Distance $PQ$ along the parallel of latitude $60^\circ\text{N}$ in nautical miles

Difference in longitude between $40^\circ\text{E}$ and $20^\circ\text{W}$ (opposite hemispheres): $$\theta = 40^\circ + 20^\circ = 60^\circ$$ Along a parallel of latitude, distance in nautical miles is: $$D = 60 \times \theta \times \cos(\text{latitude})$$ $$D = 60 \times 60 \times \cos(60^\circ)$$ $$\cos(60^\circ) = 0.5$$ $$D = 3600 \times 0.5 = 1800\text{ nm}$$ Final Answer: $1800\text{ nm}$ [3 Marks]

Part (d): Aeroplane speed in knots

$$\text{Speed} = \frac{\text{Distance (nm)}}{\text{Time (hours)}} = \frac{1800\text{ nm}}{5\text{ h}} = 360\text{ knots}$$ Final Answer: $360\text{ knots}$ [2 Marks]

7. Top 4 Costly Examination Blunders

  • Multiplying Nautical Miles by $\frac{2\pi R}{360}$ Instead of 60: On great circles, $1^\circ$ is exactly $60\text{ nm}$. Writing $\frac{\theta}{360} \times 2\pi(3437)$ gives the same result but takes five times longer. Always use $D = 60 \theta\text{ nm}$!
  • Forgetting $\cos \alpha$ on Small Circles: Forgetting to include $\cos(\text{latitude})$ when calculating distances between different longitudes along a parallel of latitude.
  • Incorrect Hemisphere Addition/Subtraction: Subtracting $40^\circ\text{E}$ and $20^\circ\text{W}$ to get $20^\circ$ instead of recognizing they are on opposite sides of Greenwich and must be added ($60^\circ$).
  • Giving Knots as km/h: Confusing nautical miles with kilometres. Knots must always be calculated using distance in nautical miles divided by hours.

8. Frequently Asked Questions

Why is 1 nautical mile equal to 1 minute of arc along a meridian?

By international historical definition, 1 nautical mile is the distance subtended by an angle of one minute ($1' = \frac{1}{60}^\circ$) at the Earth's center. Therefore, 1 degree ($60'$) of latitude equals exactly 60 nautical miles.

What is the shortest distance between two points on Earth?

The shortest route between any two locations on a sphere is along the arc of the Great Circle passing through both points (known as a great-circle route or orthodromic track).

Can longitudes be greater than 180 degrees?

No. Longitudes are measured from $0^\circ$ up to a maximum of $180^\circ\text{E}$ or $180^\circ\text{W}$, meeting at the International Date Line.

Related Senior Mathematics Revision Guides