We just saw how the lines of longitude (lines running from north-south that measure angular distance from the Prime Meridian) and latitude (lines running east-west that measure distance from the Equator) can be used as reference points.
Angular distance
The angular distance of two points can be found if you know their coordinates. Here we will focus on finding the angular distance of two points that lie on the same longitudinal line. By making this restriction, we are able to visualise the angular distance using a 2D representation or diagram. Finding the angular distance between ANY two points is a harder concept that is not required for this course.
Have a look at these two different cases.
| Case 1 | Case 2 |
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| Where the two positions are either BOTH north or BOTH south of the Equator. | Where the two positions are in different hemispheres, (one north and one south) |
From these diagrams we can see what we will need to calculate when asked to find the angular distance between A and B.
| Case 1 | Case 2 |
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Because both positions lie in the same hemisphere the distance between the two positions (the angle marked in BLUE) will be the difference between the latitude of A and the latitude of B.
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Because both positions are in different hemispheres, the distance between the two positions will be the sum of the latitude of A and the latitude of B.
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Worked examples
Example 1
Find the angular distance between the points on Earth with coordinates of ($32°$32°$N$N, $55°$55°$E$E) and ($29°$29°$N$N, $55°$55°$E$E).
By drawing a quick sketch of the location of these points we can see that
- they both lie on the same longitudinal line, (measuring $55°$55°$E$E)
- they are both north of the Equator
- we can see the angle of $29°$29°, and $32°$32° and can work out that the angle between them is the difference between $32°$32° and $29°$29°. Which is $3°$3°
Practice question
Question 1
Find the angular distance between Bali ($8^\circ$8°S $115^\circ$115°E) and Mandurah ($33^\circ$33°S $115^\circ$115°E) correct to the nearest degree.
Finding distance
The diagram below shows a sector of a circle. It has been formed by an angle of size $\theta$θ at the centre and has an arc length (the curved part of the perimeter) of length $l$l.
A sector of a circle with arc length $l$l, formed by an angle of $\theta$θ.
We know that the circumference of a whole circle is given by the formula $C=2\pi r$C=2πr
To find an arc length we are finding a fraction of a whole circle. The fraction is equal to $\frac{\theta}{360^\circ}$θ360°
Therefore to find an arc length we multiply by the fraction of the circle.
The formula is $l=2\pi r\times\frac{\theta}{360^\circ}$l=2πr×θ360°.
If you draw a circle around a sphere, like a tennis ball, you can draw circles of all different sizes. The biggest circles you can draw around the sphere are called a great circle. All of the other circles are called small circles.
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On the Earth, all lines of longitude form half a great circle. The Equator is also a great circle. All lines of latitude, except for the equator, are small circles. There are many other great circles and small circles that can be drawn on the Earth. The centre of a great circle is the same as the centre of the sphere.
The shortest distance between two points will be the arc length of the great circle that joins the points.
When finding the distance between two positions on the Earth's surface, we will use the arc length formula with the angle $\theta$θ being the angular distance as described above, and will use the radius of Earth which is approximately $6400$6400 km. Since we will be finding the arc length of a great circle with its centre aligned with the centre of the Earth.
$\text{Distance on Earth}=2\times\pi\times6400\times\frac{\text{angular distance}}{360^\circ}$Distance on Earth=2×π×6400×angular distance360°
Practice questions
Question 2
Consider the two points with coordinates of ($39$39°$N$N, $121$121°$E$E) and ($24$24°$S$S, $121$121°$E$E).
Find the angular distance between them.
Using the approximation that the radius of the earth is $6400$6400 km, find the distance between them to the nearest kilometre.