Surface Area of a Tree

A tree grows strong from adversity

Unknown.

Determining the Surface Area of a Tree

OUTCOMES – Measurement and Geometry

Mathematics 8

M01 Students will be expected to develop and apply the Pythagorean Theorem to solve problems.

M03 Students will be expected to determine the surface area of right rectangular prisms, right triangular prisms, and right cylinders to solve problems.

Mathematics 9

SCO G01 Students will be expected to determine the surface area of composite 3-D objects to solve problems. [C, CN, PS, R, V] (Communication, Connections, Problem Solving, Reasoning, Visualization)

JOB CONNECTIONS (briefly tell students all the job opportunities available for “forest geometry” – have the jobs posted on a classroom bulletin board for reference throughout the unit)

1. Forestry and Timber Industry

Foresters and Forest Managers: Calculate "basal area" (the cross-sectional surface area of a tree trunk at breast height) to determine forest density and schedule necessary thinning operations.
Timber Cruisers: Use surface measurements, such as diameter at breast height (DBH) and tree height, to estimate the gross volume of wood and its commercial value for harvesting.
Silviculture Technicians: Monitor the growth of seedlings through surface measurements (morphological measurements) to ensure healthy reforestation.

2. Ecology and Climate Science

Ecologists: Study total surface area (leaf area index and bark surface) to understand carbon sequestration rates, metabolic scaling, and how much CO₂ a forest can store.
Climate Researchers: Use tree surface data to model energy exchange, water budgets, and mineral cycles between forests and the atmosphere.
Hydrologists: Calculate how much rain is intercepted by tree canopies versus how much reaches the ground (stem flow) to predict surface runoff and flood risks.

3. Urban Planning and Maintenance

Arborists: Assess the health and structural integrity of individual trees. They may need to know the surface area for proper application of treatments like spraying pesticides or fungicides.
Landscape Architects and Urban Foresters: Design urban spaces using tree leaf area density (LAD) to predict cooling effects on buildings and overall thermal environments in cities.
Utility Arborists: Measure tree spread and surface to manage vegetation proximity to power lines and prevent outages.

4. Biological Research

Entomologists and Microbiologists: Study bark and leaf surface areas to determine the habitat capacity for various insects, lichens, and algae.
Botany Researchers: Investigating how species-specific architectural traits, like branching patterns, influence the total surface area and bifurcation frequency.

5. Technology and Engineering

Computer Vision Engineers: Develop 3D modeling tools (like LiDAR) that automatically segment and measure tree surface area from point-cloud data for environmental monitoring.
Geospatial Analysts: Use LiDAR and photogrammetry to provide large-scale resource analytics to government agencies and private forestry clients.

PREVIOUS KNOWLEDGE

Pythagorean theorem The Pythagorean theorem is a fundamental rule in geometry stating that in any right-angled triangle, the square of the hypotenuse (the longest side, 'c') is equal to the sum of the squares of the other two sides ('a' and 'b'), expressed as the equation a² + b² = c²; it allows you to find the length of any missing side if you know the other two.

Surface area formulas for 3D objects: cylinder A=2πrh+2πr2 sphere – A=4πr2

OPENING

Gratitude to nature opening

LESSON

Part I

Explain to students we are going to figure out the surface area of a tree. Explain because we are using a real-life object the final math answer will need to be a best estimate due to the irregular shape of a tree.

Have students determine which two shapes make up a tree. Cylinder is the trunk and sphere are the branches of the top of the tree which is called the canopy, which is as best as a sphere as can be. Think of a child’s drawing of a tree.

Cylinder

With all students, use the sewing measuring tape to determine the circumference of the tree trunk.

Next have students decide how best to determine the height of the tree to where it branches out. Place various measuring devices out for the students: compass, laser distance reader, sewing measuring tape, carpenter’s measuring tape, trundle wheel. Step back and observe and assess the students’ ability to work out the problem. Wait three minutes then if no one has solved it do it together as a class.

To work out the height of the tree (cylinder) have students use the Pythagorean Theorem. Choose one tree to measure its height with a metre stick. Have one student:

Hold the metre stick horizontally. Place one end of the stick near your eye (be careful), fully extend your arm out grasping the stick, this place in now “the end of the stick”
Slowly flip the stick to a vertical position not adjusting the position of where your hand is located on the stick (from step 1), keep your arm at shoulder height
Visually line up the top of stick to the top of the tree, using one eye does help. To do this you will need to adjust your standing position until the top of the stick matches the top of the tree and the bottom of the stick matches the base of the tree. This is your standing position.
Measure the length of your standing position to the tree. You can measure by strides (if you know your stride pace) or use a trundle wheel or carpenters 100 yard measuring tool. If the distance from the standing position to the tree is 10 metres, then the tree is 10 metres tall.

Can show these two videos inside first or after the lesson as a review lesson:

https://www.youtube.com/watch?v=cDy5OjfMfZ8 OR

https://www.youtube.com/watch?v=vHptCAgcl-4

Part II

As a group try to determine the radius of the top of the tree (sphere)

Determine the diameter (2R)

Problem solve with the students: look at its shadow the sphere makes on the ground or estimate measuring one side of the sphere to the other while standing on the ground. Measure with a measuring tape divide the answer by 2 is the radius. Then determine the surface area by using this formula A=4πr2. Remind students this is a best estimate.

Part III

Add the surface area of the cylinder and the sphere to determine the surface area of the tree.

CLOSING

Have students gather in a circle. Have students turn to the person next to them. Have them do a think, pair, share with each other to determine what they learned from this lesson. Ask for anyone to share something they learned from this lesson.

Have students hold open their arms to close with the saying, “All my relations”. This symbolizes the Indigenous belief all living things including humans, plants, animals, and insects have an interdependent relationship.