Latitude, longitude and great-circle distances (QCE General Mathematics Unit 3)

What this dot point is asking

This is the first sub-topic, "Locations on the Earth", of Unit 3 Topic 5: Earth geometry and time zones in the QCAA General Mathematics 2025 syllabus. You must:

• understand the meaning of great circles;
• understand angles of latitude and longitude, in decimal degrees and in degrees and minutes;
• locate and state positions on Earth's surface;
• calculate angular distance (in degrees and minutes) and distance (in kilometres) between two places on the same meridian, using $D = 111.2 \times \text{angular distance}$;
• calculate angular distance and distance between two places on the same parallel of latitude, using $D = 111.2 \cos\theta \times \text{angular distance}$, where $\theta$ is the latitude of the parallel;
• solve practical problems involving latitude, longitude, angular distance and distance.

The syllabus builds the constant 111.2 (kilometres per degree along a great circle) into both formulas, so you never need the Earth's radius itself: use 111.2 exactly as written. The second sub-topic, time zones, is on time zones and time differences.

The answer

Great circles and small circles

A great circle is a circle on the Earth's surface whose centre is the centre of the Earth. It cuts the Earth into two equal halves and has the largest possible radius, the radius of the Earth. The equator and every meridian (a line of longitude running from pole to pole, together with the meridian on the opposite side of the Earth) are great circles.

A small circle is a circle on the surface whose centre is not the centre of the Earth. Every parallel of latitude except the equator is a small circle, and parallels get smaller towards the poles.

The shortest distance between two points on the surface of a sphere lies along the great circle through them. That is why aircraft on long routes follow great-circle paths, which look curved on a flat map.

Latitude and longitude

• Latitude is the angle north (N) or south (S) of the equator, measured at the centre of the Earth, from $0\degree$ at the equator to $90\degree$ at the poles. Lines of equal latitude are parallels.
• Longitude is the angle east (E) or west (W) of the prime meridian, which passes through Greenwich in London, from $0\degree$ to $180\degree$. Lines of equal longitude are meridians. The meridian $180\degree$ is roughly followed by the International Date Line.

A position is stated latitude first, then longitude, each with its direction: Brisbane is at about $27.5\degree$S, $153.0\degree$E. To locate a position on a map, globe or GPS app, find the parallel for the latitude, then move along it to the meridian for the longitude. On most maps, latitude values are printed on the left and right edges and longitude values along the top and bottom.

Degrees and minutes. One degree is divided into 60 minutes ($1\degree = 60'$).

• Minutes to decimal degrees: divide by 60. $23\degree 15' = 23 + \frac{15}{60} = 23.25\degree$.
• Decimal degrees to minutes: multiply the decimal part by 60. $18.75\degree = 18\degree + 0.75 \times 60' = 18\degree 45'$.
• When adding or subtracting in degrees and minutes, carry or borrow 60 minutes, just as you would with hours and minutes: $27\degree 28' - 16\degree 55' = 26\degree 88' - 16\degree 55' = 10\degree 33'$.

Most scientific calculators have a degrees-minutes-seconds key that performs these conversions; show the conversion in your working anyway.

Where 111.2 and $\cos\theta$ come from

A great circle has the Earth's radius, about 6371 km, so its circumference is $2\pi \times 6371 \approx 40\,030$ km. Dividing by $360\degree$ gives about $111.2$ km for each degree of angle at the centre:

A parallel at latitude $\theta$ has radius $R\cos\theta$ (the figure shows the right-angled triangle that gives this), so its circumference, and the length of each of its degrees, is multiplied by $\cos\theta$:

At the equator $\cos 0\degree = 1$ and the formulas agree. At $60\degree$, $\cos 60\degree = 0.5$, so a degree of longitude is only 55.6 km. At the poles, $\cos 90\degree = 0$ and every meridian meets at a point.

Finding the angular distance

Same meridian (north-south). Use the latitudes.

• Same hemisphere: subtract. $35\degree$N and $20\degree$N are $15\degree$ apart.
• Opposite sides of the equator: add. $12\degree$N and $25\degree$S are $37\degree$ apart.

Same parallel (east-west). Use the longitudes.

• Both east or both west: subtract. $115\degree$E and $153\degree$E are $38\degree$ apart.
• One east and one west: add. $20\degree$W and $25\degree$E are $45\degree$ apart. If the sum is more than $180\degree$, the shorter way round crosses the $180\degree$ meridian: use $360\degree$ minus the sum.

Comparing the two formulas

The further a parallel is from the equator, the shorter each degree of longitude becomes, reaching zero at the poles.

A parallel is not the shortest route

Two stations on $60\degree$S at longitudes $0\degree$ and $180\degree$ are $111.2 \cos 60\degree \times 180 = 10\,008$ km apart along the parallel. But the meridians $0\degree$ and $180\degree$ together form a single great circle through the South Pole, and along it the stations are only $30\degree + 30\degree = 60\degree$ apart: $111.2 \times 60 = 6672$ km. The great-circle route saves 3336 km. The syllabus only asks you to calculate distances along a meridian or a parallel, but this example shows why "great circle" and "shortest distance" go together.