8.02 Distance between two places on Earth

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 from a reference point can be found if you know their coordinates.  Here we will focus on finding the angular distance from the centre of the Earth for 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 from the centre of the Earth between ANY two points is a harder concept that is not needed for this course. 

Have a look at these two different cases. 

Case 1	Case 2
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
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.	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.

 

Worked examples

Example 1

Find the angular distance between the points on Earth with coordinates of ($32^\circ$32°$N$N, $55^\circ$55°$E$E) and ($29^\circ$29°$N$N, $55^\circ$55°$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^\circ$55°$E$E)
• they are both north of the Equator
• we can see the angle of $29^\circ$29°, and $32^\circ$32° and can work out that the angle between them is the difference between $32^\circ$32° and $29^\circ$29°.  Which is $3^\circ$3°
   

 

 

Practice question 

Question 1

 

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.

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:

 $l=\frac{\theta}{360^\circ}\times2\pi r$l=θ360°​×2πr

Great and small circles

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.

 

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. Move the points in the following applet to see the plane that divides the sphere in two and boundary forms a great circle through the two given points with a shared centre with the sphere.

Created with Geogebra

Practice question

question 2

Distance between two points with the same longitude

When finding the distance between two positions on the Earth's surface, we will use the arc length formula, $l=\frac{\theta}{360^\circ}2\pi r$l=θ360°​2πr, with the angle $\theta$θ being the angular distance from the centre of the Earth, as we described above. And we will use the radius of Earth which is approximately $6371$6371 km, since we will be finding the arc length of a great circle with its centre aligned with the centre of the Earth.

So the formula to find the distance between two points with the same longitude (or indeed two points on the equator) is:

$\text{Distance on Earth}=\frac{\text{Angular distance}}{360^\circ}\times2\pi\times6371$Distance on Earth=Angular distance360°​×2π×6371

We can simplify this formula by evaluating the terms which do not change. Since, $\frac{2\pi\times6371}{360}\approx111.2$2π×6371360​≈111.2, we have:

$\text{Distance on Earth}\approx111.2\times\text{Angular distance}$Distance on Earth≈111.2×Angular distance

 

Note: Sometimes slightly different approximations for the Earth's radius are used such as $6400$6400 km.

Practice questions

Question 3

Question 4

 

Small circle distance between two points with the same latitude 

We can find the distance travelled along a parallel between two points with the same latitude if we know the radius of the small circle formed by the parallel of latitude. Let's look at how we can find this radius:

 

We want to find the radius $r$r, the length $AB$AB, of a parallel located at $\theta$θ. Notice the line segment $BC$BC will be the radius of the Earth and the angle $ABC$ABC will be equal to $\theta$θ due to alternate angles in parallel lines. Let's look closer at the triangle:

 

Using trigonometry we can create a formula for $r$r:

$\cos\theta$cosθ	$=$=	$\frac{\text{Adjacent}}{\text{Hypotenuse}}$AdjacentHypotenuse​
	$=$=	$\frac{r}{6371}$r6371​
$\therefore\ r$∴ r	$=$=	$6371\cos\theta$6371cosθ

 

 

Then using the angular distance we can use the arc length formula once again with this new radius. In this case the angular distance will be measured as the angle between the points relative to the centre of the small circle of the parallel on which they are located. This is calculated similarly to the angular distance between points with the same meridian but this time we focus on the distance between the longitude values.

So the formula to find the distance along a parallel between two points at the same latitude is:

$\text{Distance on Earth}=\frac{\text{Angular distance}}{360^\circ}\times2\pi\times6371\cos\theta$Distance on Earth=Angular distance360°​×2π×6371cosθ

Which again we can simplify this formula by evaluating the terms which do not change. Since, $\frac{2\pi\times6371}{360}\approx111.2$2π×6371360​≈111.2, we have:

$\text{Distance on Earth}\approx111.2\times\cos\theta\times\text{Angular distance}$Distance on Earth≈111.2×cosθ×Angular distance

 

Note: This is not the shortest distance between the two points, that would be the great circle distance between the two points. This is the distance along the line of latitude between the two points.

 

Worked example

Example 2

A plane travels from Sydney ($34^\circ S$34°S, $151^\circ E$151°E) to Cape Town($34^\circ S$34°S, $18^\circ$18°).

(a) Are the points located on the same meridian or the same parallel?

Think: Meridians coincide with points of the same longitude and parallels coincide with points of the same latitude.

Do: They both lie on the same latitudinal line at $34^\circ S$34°S and are hence located on the same parallel.

(b) How many degrees east of Cape Town is Sydney?

Think: Both cities are east of the Prime Meridian so we can find the difference between their longitude values to find how much further east Sydney is from Capetown. This will be the angular distance relative to the centre of the small circle formed by the parallel at $34^\circ S$34°S.

Do: 

$\text{Angular distance}$Angular distance	$=$=	$151^\circ-18^\circ$151°−18°
	$=$=	$133^\circ$133°

(c) If a plane travels along the parallel from Sydney to Cape Town approximately how far will it travel?

Think: We can use the formula $\text{Distance on Earth}\approx111.2\times\cos\theta\times\text{Angular distance}$Distance on Earth≈111.2×cosθ×Angular distance, with $\theta=34^\circ$θ=34° and the angular distance we calculated in part (b).

Do:

$\text{Distance on Earth}$Distance on Earth	$\approx$≈	$111.2\times\cos\theta\times\text{Angular distance}$111.2×cosθ×Angular distance
	$=$=	$111.2\times\cos\left(34^\circ\right)\times133$111.2×cos(34°)×133
	$=$=	$12261$12261 km

The plane will fly approximately $12261$12261 km.

Reflect: This distance does not take into account the height of the plane - this is the distance along the surface of the sphere. It is also not the shortest distance between the two points. If we search online for the distance between Cape Town and Sydney the distance of the shortest path along the great circle of $11005$11005 km will be given. Have you ever noticed the curved paths planes take on tracking maps?

The shortest distance between any two general locations requires further steps to find the angular distance relative to the centre of the Earth. This is beyond the scope of the course but we can use online calculators and map services to calculate such distances given the latitude and longitude of a location.

 

 

Practice questions

Question 5

Question 6