How Would you Measure the Height of these Trees with No Equipment?

FOR 274: Forest Measurements and Inventory

Trigonometry, Geometry and Rates

• Reminder of SOH, CAH, TOA

• Areas and Volumes

• Rates in Forestry

• Reporting Data

Forest Measurements: A World of Triangles

A large amount of forest measurements uses the mathematics of Triangles:

Forest Measurements: A World of Triangles

A large amount of forest measurements uses triangles

To help us take easier (or fewer) measurements we need to know as many mathematical tricks as possible.

Triangles: Remembering Angles

This simple proof shows that the 3 angles in a triangle add up to 180°.

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Triangles: Right Angled Triangles

Remembering SOH CAH TOA:

Sin θ = Opposite / Hypotenuse (SOH)Cos θ = Adjacent / Hypotenuse (CAH)Tan θ = Opposite / Adjacent (TOA))

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Triangles: Pythagoras

Remembering Pythagoras:

a2 + b2 = c2

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Triangles: The Sine Rule

Used to find the lengths of each side and all its angles when we know either a) two angles and one side, or b) two sides and an opposite angle.

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Triangles: The Cosine Rule

Used to find the angles of a triangle when the lengths of each side are known:

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Forest Measurements: Areas and Volumes

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A considerable amount of forestry deals with calculating the surface area and volume of plants & landscape features

• Surface area to volume ratio is a very important parameter in assessing fire fuels

• Log volumes are very important in evaluating timber value for sales

• Crown area and volume is very important in evaluating crown fire spread models

• Cross-sectional channel area and volume of flow is important watershed hydrology and plant ecophysiology

• Cross-sectional area of stems (basal area) is very important is modeling forest growth

Log Volumes: Geometric Solids

Geometric tree shapes follow the equation Y = K√Xr, where r =0, 1, 2, 3, etc

Log Volumes: Geometric Solids

Logs are not perfect cylinders!

Logs taper from one end to another

Truncated sections of a tree can be approximated as geometric shapes:- Cone- Paraboloid- Neiloid

Log Volumes: Geometric Solids

Volume of any geometric soild= “average cross-sectional area” * Length

Huber’s Cu Volume = (B1/2)*L

Smalian’s Cu Volume = (B+b)/2 * L

Newton’s Cu Volume = (B+4B1/2+b)/6*L

Log Volumes: Geometric Solids

Forest Measurements: Areas and Volumes

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Circle Area = π*r2

Cylinder Area = 2*(π*r2) + 2(π*r*h)

Cone Area = π*r2 + π*r*s

Cylinder Volume = π*r2*h

Cone Volume = 1/3 π*r2*h

Question: If we measure the diameter of a tree in Inches, how would we convert this into square feet?

Forest Measurements: Flow Rates

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A considerable amount of forestry deals with flow rates and the “gradient” equations

• Xylem flow and phloem flow is important in plant growth

• Flow of water and nutrients through plants and soils

Forest Measurements: Flow Rates

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A considerable amount of forestry deals with flow rates and the “gradient” equations

• Flow of water and sediments in Forest watersheds

Rates in Forestry: Gradient Equations

A gradient tells us how much a certain parameter changes with distance. All gradient equations used in forestry have the form:

Change (flux) = -Constant * (Quantity 2 – Quantity 1)

Quantity 2 is always greater (higher potential, height, etc)

Diffusion: The movement of high concentration (c) to low concentration

Change = -D*(c2 – c1)/(position of c2 – position of c1)

Conduction: The movement of high temperature (T) to low temperature

Change = -k*(T2 – T1)

Rates in Forestry: Plant Growth

Growth Curve: S- (or sigmoid) shaped and shows cumulative growth at any age

Height

Age

How do we calculate the instantaneous rate of change?

Rates in Forestry: Plant Growth

Rate of Growth Curve: Rapid growth in youth with decreasing rate as tree matures

Rate

Age

Rates in Forestry: Plant Growth

Current and mean annual growth curves:

DBH Growth

Age

Rates in Forestry: Plant Growth

A measure of the average rate of change in size or volume over a given time interval

Growth percent = 100 * (V2-V1)/(N*V1)

V1 = Volume or size at start

V2 = Volume or size at end

N = number of years

This measure is analogous to interest rates as found in economics

Forest Measurements: Reporting Data

Forestry is a professional career. As such you will be preparing technical reports.

A common mistake by new foresters is to make reports to short. The problem is that in many cases the report is the only concrete evidence that any work was done.

To be recognized as a competent professional the ability to wrote grammatically correct and mathematically accurate reports is critical

Forest Measurements: Reporting Data

Main Elements of a Report

- Title Page

- Table of Contents

- Introduction

- Review of Prior Work

- Study Area Description

- Collection of Field Data and Other Analysis

- Analysis of Results

- Summary and Conclusions

- Literature Cited

Forest Measurements: Proper Use of Tables

Forest Measurements: Proper Use of Figures

Forest Measurements: Proper Use of Graphs

Message of the Day: Understanding how to use trigonometry, geometry, and rates are essential in professional forestry.