Earth Geometry is a full 12-mark structured question in Section B of ECZ Grade 12 Mathematics Paper 2. Candidates frequently lose marks on navigational sub-questions involving flight time, speed in knots, and shortest distances over the poles. This tutorial provides clear step-by-step formulas and fully solved exam problems.
Paper 2 Master Strategy:
Earth Geometry questions follow a predictable pattern: (a) Position coordinates, (b) Great circle distance in nm or km, (c) Small circle distance along latitude, and (d) Speed / Flight time calculation in knots!
1. Navigation Units & Formulas
Before solving past paper questions, memorize these core definitions:
Essential Formulas:
- Nautical Mile (nm): $1^\circ$ of arc along a Great Circle = $60\text{ nautical miles (nm)}$.
- Distance along Great Circle (Meridian / Equator):
$$D_{\text{GC}} = \theta \times 60\text{ nm} \quad \text{or} \quad D_{\text{GC}} = \frac{\theta}{360^\circ} \times 2 \pi R \quad (R \approx 6370\text{ km})$$ - Distance along Small Circle (Parallel of Latitude $\alpha$):
$$D_{\text{SC}} = \theta \times 60 \cos \alpha\text{ nm} \quad \text{or} \quad D_{\text{SC}} = \frac{\theta}{360^\circ} \times 2 \pi R \cos \alpha$$ - Speed in Knots:
$$\text{Speed (knots)} = \frac{\text{Distance (nautical miles)}}{\text{Time (hours)}}$$
2. Worked Exam Example: Speed & Flight Time
Worked Example 1 (Flight Time & Knots):
An aircraft flies due East from point $A(60^\circ\text{N}, 20^\circ\text{W})$ to point $B(60^\circ\text{N}, 40^\circ\text{E})$ along the parallel of latitude $60^\circ\text{N}$.
(a) Find the difference in longitude ($\theta$) between A and B.
Since A is West and B is East across the Prime Meridian:
$$\theta = 20^\circ + 40^\circ = 60^\circ$$
(b) Calculate the distance AB along the parallel of latitude in nautical miles.
Using $D_{\text{SC}} = \theta \times 60 \cos \alpha$ with $\alpha = 60^\circ$ and $\theta = 60^\circ$:
$$D_{\text{AB}} = 60 \times 60 \times \cos(60^\circ) = 3600 \times 0.5 = 1800\text{ nm}$$
(c) If the aircraft flies at a constant speed of $450\text{ knots}$, calculate the flight time in hours.
$$\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{1800\text{ nm}}{450\text{ knots}} = 4\text{ hours}$$
(d) If the aircraft departs point A at 08:00 hours local time, find its arrival local time at point B.
Time difference due to longitude: $\frac{60^\circ}{15^\circ/\text{hr}} = 4\text{ hours}$ ahead (East is ahead).
$$\text{Arrival Local Time} = \text{Departure Time} + \text{Flight Duration} + \text{Longitude Time Gain}$$
$$\text{Arrival Time} = 08:00 + 4\text{ hrs} + 4\text{ hrs} = 16:00\text{ hours}$$
Teacher Tip:
Remind candidates to distinguish between flight duration (elapsed travel time) and local arrival time (which accounts for timezone shifts across meridians)!
3. Shortest Distance over the North/South Pole
When two places lie on opposite meridians (sum of longitudes $= 180^\circ$), the shortest distance between them is along the Great Circle route passing directly over the Pole.
Worked Example 2 (Polar Route):
Point $P$ is $(70^\circ\text{N}, 30^\circ\text{E})$ and point $Q$ is $(50^\circ\text{N}, 150^\circ\text{W})$.
Notice that $30^\circ + 150^\circ = 180^\circ$, meaning P and Q are on opposite sides of the Earth.
Angular distance over the North Pole ($\theta$):
$$\theta = (90^\circ - 70^\circ) + (90^\circ - 50^\circ) = 20^\circ + 40^\circ = 60^\circ$$
Shortest Distance PQ in nautical miles:
$$D_{\text{polar}} = \theta \times 60\text{ nm} = 60 \times 60 = 3600\text{ nm}$$
Scientific Calculator Setup:
Ensure your scientific calculator is set to DEGREE (D) mode before calculating $\cos \alpha$. If set to RAD or GRAD mode, all trig calculations will be incorrect!
4. Summary of Exam Pitfalls
- Confusing $\cos \alpha$ placement: Using $\cos \alpha$ when calculating Great Circle distance (meridian) instead of Small Circle distance (latitude).
- Wrong $\pi$ value: Using $\pi = 3.14$ when the ECZ question explicitly specifies $\pi = 3.142$ or using the calculator $\pi$ button. Always check the instruction on page 1 of the paper!
- Forgetting units: Writing $1800$ instead of $1800\text{ nm}$ or $1800\text{ km}$.