QCE General Mathematics - Unit 3 - Earth geometry and time zones
Locations on the Earth | QCE General Mathematics
Learn great circles, latitude, longitude, angular distance and distance along meridians or parallels in QCE General Mathematics.
Updated 2026-05-18 - 5 min read
QCAA official coverage - General Mathematics 2025 v1.3
Exact syllabus points covered
- Understand the meaning of great circles.
- Understand the meaning of angles of latitude and longitude (in decimal degrees, and degrees and minutes) in relation to the equator and the prime meridian respectively.
- Locate positions on Earth's surface given latitude and longitude, e.g. using a globe, map, GPS and other digital technologies.
- State latitude and longitude for positions on Earth's surface, e.g. investigating a map of Australia and locating boundary positions for Aboriginal peoples' and Torres Strait Islander peoples' language groups, Australian landmarks or local land boundaries.
- Calculate angular distance and distance between two places on Earth on the same meridian: $D=111.2\times\text{angular distance}$, where $D$ is distance in kilometres.
- Calculate angular distance and distance between two places on Earth on the same parallel of latitude: $D=111.2\cos\theta\times\text{angular distance}$, where $D$ is distance in kilometres and $\theta$ is latitude.
- Solve practical problems involving latitude, longitude, angular distance and distance.
Earth geometry treats Earth as a sphere so that locations can be described using latitude and longitude. This model is not perfect, but it is accurate enough for the practical distance and time-zone calculations in General Mathematics.
Original Sylligence diagram for general earth latitude longitude.
Latitude and longitude
Latitude measures angle north or south of the equator. The equator is $0^\circ$, the North Pole is $90^\circ N$ and the South Pole is $90^\circ S$.
Longitude measures angle east or west of the prime meridian. The prime meridian is $0^\circ$ and the International Date Line is near $180^\circ$.
Great circles pass through the centre of Earth. Meridians of longitude are great semicircles. The equator is a great circle. Other parallels of latitude are small circles.
Distance on the same meridian
For two places on the same meridian:
$ D=111.2 \times \text{angular distance} $
where $D$ is distance in kilometres and angular distance is in degrees.
Distance on the same parallel
For two places on the same parallel of latitude:
$ D=111.2\cos(\theta)\times \text{angular distance} $
where $\theta$ is the latitude.
Worked example
Great circles and small circles
A great circle is formed when a plane cuts through the centre of a sphere. The equator is a great circle, and meridians form great semicircles from pole to pole. Other parallels of latitude are small circles because their radius is smaller than Earth's radius.
This explains why east-west distance along a latitude is shorter as you move away from the equator.
Degrees and minutes
Latitude and longitude may be written in decimal degrees or in degrees and minutes:
$ 1^\circ=60' $
For example:
$ 27^\circ 30'=27.5^\circ $
Convert to decimal degrees before using a calculator unless the question explicitly asks for degrees and minutes.
Angular distance rules
For two latitudes in the same hemisphere, subtract. For two latitudes in opposite hemispheres, add their magnitudes.
For longitudes, be careful near $0^\circ$ and $180^\circ$. The smaller angular separation is usually used unless the question specifies a particular route. In General Mathematics, most same-parallel questions are set up so the intended angular distance is straightforward.
Depth: latitude, longitude and angular distance
Latitude measures angular distance north or south of the Equator. Longitude measures angular distance east or west of the Prime Meridian. A location such as $27^\circ 28' S, 153^\circ 02' E$ gives latitude first, then longitude.
Degrees and minutes are used because one degree is split into $60$ minutes:
$ 1^\circ=60' $
For example:
$ 27^\circ 30'=27.5^\circ $
and:
$ 153.25^\circ=153^\circ 15' $
Great circles and small circles
A great circle cuts the Earth into two equal hemispheres. The Equator and every meridian form great circles. Parallels of latitude other than the Equator are small circles because their radius is smaller than Earth's radius.
This distinction affects distance:
| Movement | Circle type | Distance rule | |---|---|---| | north-south along a meridian | great circle | $111.2$ km per degree approximately | | east-west along the Equator | great circle | $111.2$ km per degree approximately | | east-west along another parallel | small circle | distance is reduced by the latitude factor |
In many General Mathematics questions, a formula book or question statement supplies the needed Earth-radius formula. The key is deciding which angular difference is being measured.
Same meridian and same parallel problems
If two places have the same longitude, they lie on the same meridian. Their angular distance is found from the difference in latitudes, remembering signs:
- north and north: subtract the latitudes
- south and south: subtract the latitudes
- one north and one south: add the magnitudes
If two places have the same latitude, they lie on the same parallel. Their east-west angular difference is found from longitudes:
- east and east: subtract
- west and west: subtract
- one east and one west: add, unless crossing the $180^\circ$ meridian gives a shorter path
Always include whether the answer is an approximation, because Earth is modelled as a sphere.