Mathematics of bushfire

Mathematics of bushfire

Mathematical concepts and tools can be used to enhance our understanding of bushfire behaviour and impacts.

Fire in indigenous culture

Fire is crucial in Aboriginal and Torres Strait Islander culture. Fire is used by Aboriginal and Torres Strait Islander people to manage the landscape and promote biodiversity – overall, fire plays an integral part in maintaining healthy country. On the other hand, intense fires can kill sensitive plants and seedlings, and can burn into the forest canopy. When the canopy is destroyed by fire it makes it more difficult for the forest to recover. Intense fires can also destroy logs and tree-hollows, which small mammals, reptiles and birds depend upon for their survival, and lead to other problems like soil erosion. Rapidly spreading fires can also burn vast areas, leaving little time for animals (including people) to escape.

Aboriginal and Torres Strait Islander learnt to understand how different environmental factors affect how fast a fire will spread and how intensely it will burn. Fire had to be very carefully introduced into the landscape so it would not spread too fast or burn too hot. As such, fire was only used in very specific seasons, under favourable weather conditions and when certain indicator species suggested it was the best time to do so. Ignition was done very carefully, so as to produce fires that burnt in very specific patterns. Aboriginal and Torres Strait Islander people have a highly developed understanding of how fire behaves in country, and they use this knowledge to keep the country healthy and productive.

Understanding bushfire behaviour and spread

In broad terms, bushfire behaviour is influenced by three main factors: weather, fuel and terrain. Key weather variables are wind, temperature, and moisture (either in the form of rain or relative humidity). Temperature and moisture influence how dry the fuels are, with fire spreading faster and more intensely in drier fuels. The wind also determines how fast a fire will spread as well as the direction that it spreads – fires tend to propagate with the wind, and spread faster when winds are stronger. There are various mathematical equations that relate the rate of fire spread in different fuel types (e.g. forest or grasslands) to wind speed, temperature and relative humidity. There are also equations that can be used to determine the intensity of the fire, and the length and angle of the flames – generally speaking, fires spread faster in heavier fuel loads burn with greater intensity. These mathematical equations confirm the detailed knowledge of fire behaviour that has been passed from generation to generation of Aboriginal and Torres Strait Islander people over thousands of years.

The strength of the wind also determines the overall shape of a fire. Generally speaking, the winds cause a fire to spread faster in the direction of the wind and slower against the wind. This means that the fire will make more of an oval shape as it burns, with the major axis aligned with the wind. The proper name for this type of oval shape is an ellipse, which is a very important mathematical object – ellipses are also the shape the planets make as they orbit the sun. The dimensions of the ellipse can be directly related to the strength of the wind, with stronger wind producing more elongated ellipses.  Understanding the shape that fires assume allows us to use mathematical concepts to calculate the area that they burn.

Terrain also affects the rate of spread of a fire and its shape. Fires tend to spread faster when they spread up a hill, and slower when they burn downhill. In fact, it is known that a fire approximately doubles its rate of spread when a hill becomes 10 degrees steeper. So, for example, if a fire spreads at 10 km/h on flat ground, it will spread at about 20 km/h on a hill with a 10 degree slope, and about 40 km/h on a 20 degree slope. Fires burning downhill are more complicated, but can still be described by a mathematical equation.

Effects of bushfire

Aboriginal and Torres Strait Islander people understood that when country is not properly cared for, bad fires can become more likely. There is increased fuel load so these fires burn with more intensity and can destroy entire forest stands, including the tree canopies, which many birds and animals rely upon for their survival. In recent decades there have been a number of destructive wildfires that have scorched vast areas of land. These fires have killed hundreds of people and destroyed thousands of houses, they have also threatened the survival of certain plants and animals (e.g. Leadbeaters possum, corroboree frog, snow gum, sphagnum moss).

The level of fire danger on a given day can be described in terms of how hot, dry and windy the conditions are. Again, we can use a mathematical equation to tell us how dangerous a particular day is, and these ideas can be extended to allow us to compare fire danger from year to year or decade to decade. This is a very important consideration in the context of climate change – fires are expected to become more frequent, more intense and to burn greater areas into the future.

Classroom activity - Mathematics Year 9

In these classroom activities students will be introduced to some of the basic mathematical principles that underpin wildfire science, with an emphasis on how theoretical concepts are used to aid our understanding of the real world, and bushfire in particular. Students will be exposed to various algebraic relationships that link fire behavioural attributes to environmental conditions, and challenged with using these algebraic relationships to deduce information about rates of spread, flame height, flame angle and fire intensity. Through exploring these relationships, students will acquire an understanding of how algebra, trigonometry and basic functions are used in real world examples.  They will also learn to appreciate the complexity of the system that Aboriginal and Torres Strait Islander peoples have successfully managed for millennia.

Curriculum connections

This resource addresses the following content descriptions from the Australian Curriculum:

  • Express numbers in scientific notation (ACMNA210)
  • Find the distance between two points located on the Cartesian plane using a range of strategies, including graphing software (ACMNA214)
  • Calculate areas of composite shapes (ACMMG216)
  • Apply trigonometry to solve right-angled triangle problems (ACMMG224)
  • Identify everyday questions and issues involving at least one numerical and at least one categorical variable, and collect data directly and from secondary sources (ACMSP228)

This resource addresses the following excerpts from the achievement standard for Year 9 in Mathematics:

  • recognise the connections between similarity and the trigonometric ratios
  • apply the index laws to numbers and express numbers in scientific notation
  • calculate areas of shapes and the volume and surface area of right prisms and cylinders

Activity 1 – The geometry of fire

Suggested timing for activity: 30 minute discussion on concepts, theory and example problems. Two 30 minute sessions to work on assigned problems in handout

Required resources: handout with problems (see below)

  • Students are introduced to the concept of an ellipse as a generalisation of a circle. Concept of length-to-breadth ratio is introduced along with semi-major and semi-minor axes and formula for area.
  • The ellipse is linked to the basic shape of a wind-driven fire (using landsat examples) and the basic anatomy of a fire is introduced (head fire, flanks, etc). In particular, students consider the head fire as the fastest moving part of a fire.
  • Introduce the relationship between length-to-breadth ratio and wind speed via graphical handout and get students to assign wind speeds to various elliptical shapes and vice versa.
  • Get students to draw various fire shapes based on information about wind speed and direction (using Cartesian plane/compass points).
  • Estimate areas of very large fires, expressing answers using scientific notation.

Activity 2 – Fire behaviour (Part one)

Suggested timing for activity: 40 minute discussion on concepts and theory. One 40 minute session to work on problem set 1 and one 40 minute session to work on problem set 2 in handout.

Required resources: handout with problems (see below)

  • Students are introduced to the concept of head fire rate of spread, and discuss the factors that influence it (wind, temperature, relative humidity, and topographic slope both uphill and downhill).
  • Students are introduced to equations that relate rate of spread in grassland and forest to wind speed, temperature, relative humidity and topographic slope
  • Students look at Bureau of Meteorology website and find the nearest automatic weather station. Use the current data to calculate how fast a fire would spread in grassland and forest.

Activity 3 – Fire behaviour (Part two)

Suggested timing for activity: 40 minute discussion on concepts, theory and examples. One 40 minute session to work on problem set in handout. 15 minutes class discussion on the two inquiry questions listed below after watching video.

Required resources: handout with problems (see below), computer with video capability

  • Students are introduced to the concepts of flame angle, flame height and fire intensity and the additional factors that determine them (e.g. fuel load).
  • Equations for flame angle, flame height and fire intensity are introduced and discussed.
  • Students answer worksheet questions that require solving problems about rate of spread, fire intensity, flame angle, flame height. These problems require combining algebraic and trigonometric concepts.
  • Discussion/video about fire intensity and how it defines ‘good fire’ and ‘bad fire’.

For example: “Fighting carbon with fire”:

  • Inquiry-based questions:
    • Why would Aboriginal people want to avoid ‘bad fire’?
    • Reflecting back over the last two activities, why would Aboriginal people choose to ignite a fire by dropping a firestick from the top of a mountain (as opposed to the bottom)?

Activity 4 – Fire and climate

Suggested timing for activity: 40 minute discussion on concepts, theory and examples. One 30 minute session to work on problem set 1, and one 40 minute session to work on problem set 2 in handout. Problem set 2 can involve broader class discussion and students working in groups. 

Required resources: handout with problems (see below). Access to a computer with web browser.

  • Introduce the concept of fire danger rating (FDR) and describe how it is used by firefighters and to warn the community of bad fire days. Introduce classification of FDR.
  • Introduce forest fire danger index and discuss how it is calculated and how it is used in scientific studies
  • Interpreting maps/statistics about changes in fire weather and fire danger rating
    • Examine research output from an example that considers historical changes in fire weather conditions in Australia
    • Consider different ways of condensing and interpreting data
    • Interpret graphical/statistical output relating to changes in space and time
    • Make decisions based on graphical/statistical output




2 Note that because there are 365 days in a year, FFDI values above the 36th highest value are in the top 10% of all FFDI values for that year, or in other words 90% of the values are less than this value.

5 Reference: Clarke, H., Lucas, C. and Smith, P. (2013) Changes in Australian fire weather between 1973 and 2010. International Journal of Climatology33(4): 931-944.

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

The development of these resources was funded through an Australian Government initiative delivered by the University of Melbourne's Indigenous Studies Unit. The resources include the views, opinions and representations of third parties, and do not represent the views of the Australian Government. They have been developed as a proof of concept to progress the inclusion of Aboriginal and Torres Strait Islander content in Australian classrooms. In drawing on the material, users should consider the relevance and suitability to their particular circumstances and purposes.