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Lesson 4, Topic 3
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# Calculating Distance Along Great Circle

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Note that the shortest distance between two points on the surface of the earth lies on the arc length of a great circle connecting the two points. Let the arc length be $$\scriptsize \bar{AB}$$, then

$$\scriptsize \bar{AB} = \normalsize \frac{θ°}{360°} \scriptsize \times 2 \pi R$$

Where  θ° = angular difference

R = radius of the earth = 6,400Km

π = $$\frac{22}{7}$$

Example 1

A and B are two places on the earth’s surface on the same meridian. B has Latitude 20o N and A is a point north of B such that the distance AB measured along the meridian is 800Km. Calculate the Latitude of A correct to the nearest degree, (Take π = $$\frac{22}{7}$$ and R = 6,400Km.)

AB = 800 =  $$\frac{θ}{360} \; \times \; \scriptsize 2 \; \times \; \normalsize \frac{22}{7} \scriptsize \; \times \; 6400$$

θ = $$\frac{800 \; \times \; 360\; \times \; 7}{44 \; \times \; 6400}$$

= $$\frac{45 \; \times \; 7}{44}$$

θ = 7.1590

θ = 7° (nearest degrees)

Since the two points are in the same direction

θ = Latitude A – Latitude B

i.e. 7 = Latitude A – 20

Therefore, Latitude A = 7 + 20

i.e. Latitude A = 27o N

Example 2

(a) An aeroplane flies from city A (0o, 152o E) to city B (0o, 171o W), find the shortest distance travelled to reach city B

(b) How far is city B from the North Pole? (Take π = $$\frac{22}{7}$$ and R = 6,400Km.)

Solution:

The angular difference between points A and B is 323o which gives the major segment – it does not give the shortest distance, rather the minor segment of 37o gives the shortest distance.

i.e. $$\scriptsize \bar{AB}\; = \normalsize \frac{θ}{360} \scriptsize \; \times \; 2 \; \times \; \normalsize \frac{22}{7} \scriptsize \; \times \; 6400$$

$$= \frac{θ}{360} \; \times \; \scriptsize 2 \; \times \; \normalsize \frac{22}{7} \scriptsize \; \times \; 6400$$

= 4,134.60317

i.e.     AB = 4,135Km

Recall that from the equator 0o to the North Pole, the angular difference is 90o.

Therefore, $$\scriptsize \bar{BN} = \normalsize \frac{θ}{360} \; \times \; \scriptsize 2 \; \times \; \normalsize \frac{22}{7} \scriptsize \; \times \; 6400$$

= $$\scriptsize \bar{BN} = \normalsize \frac{90}{360} \; \times \; \scriptsize 2 \; \times \; \normalsize \frac{22}{7} \scriptsize \; \times \; 6400$$

=10,057.1429

$$\scriptsize \bar{BN} = 10,057Km$$

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