How do you calculate the cross-section of a single tree at breast height?

The cross-sectional area at breast height is the area of a circle calculated from the diameter at breast height. It is therefore determined by multiplying π by the square of the value obtained when the diameter (DBH) is divided by 2. If the diameter is given in inches and the result is to be expressed in square feet, then the simplified formula BA = 0.005454 × (DBH)² can be used. If the diameter is given in centimeters and the result is to be expressed in square meters, then the formula BA = 0.00007854 × (DBH)² can be used. The cross-sectional area of a tree with a diameter of 14 inches would be approximately 1.07 square feet.

How to Calculate Tree Cross-Sectional Area per Acre?

Measure the diameter at breast height (DBH) of each tree within a representative study area, calculate the corresponding cross-sectional area for each tree, add up those values and divide by the area of the study site in acres. For example, if six trees have a total cross-sectional area of 4.28 square feet on a 900-square-foot study plot, then you would divide 4.28 by 0.0207 to get about 207 square feet per acre.

What is the Prism Method or Angle Measurement Method?

This is a quick method that can be used in the field and is called point sampling. You stand at a point and rotate a wedge prism or angle gauge around you to scan the area. Trees that appear wider than the movement of the prism are counted. When this number is multiplied by the cross-sectional-area factor for the instrument, it gives directly the cross-sectional area at breast height per acre or hectare without measuring diameter. By averaging counts from several points, accuracy of estimation can be improved.

Where is height measured at bust level?

In many regions of the world including Australia, Canada and Europe, height is measured at breast height which is approximately 4.3 feet above ground level or 1.3 meters. In the United States, the measurement position for height at breast height is 4.5 feet or 1.37 meters. For slopes, diameter is measured from the uphill side of the tree.

Why is the cross-sectional area at breast height important for forestry?

The cross-sectional area at breast height gives an idea of how much space a tree occupies and allows for the stand density and competition to be measured with one number. Managers use this to assess whether thinning is required in a forest stand, simulate growth, estimate timber volume and biomass, and represent habitat condition. This is because crown cover is closely related to cross-sectional area at breast height.

Basal Area Calculator

Basal area

What would you like to calculate?

Find the basal area of a single tree, roll a plot of measured trees up into a stand density, or estimate density in the field with a prism or angle gauge.

One tree

Whole stand (measure a plot)

Prism / angle gauge

Basal area is the cross-section of a tree trunk at breast height. Enter a single tree, roll up a plot of measured trees into a stand density, or use a prism to estimate density in the field.

Single tree

Diameter at breast height (DBH)

Basal area of the tree

Please input values into the fields

Notes and chart

Breast height is 1.3 m (about 4.3 ft) above the ground in much of the world, and 4.5 ft (1.37 m) in the United States. Measure the diameter there, on the uphill side for a tree on a slope.

The breast height diameter is the cross-sectional area of a tree trunk measured at breast height. In forestry, the breast heights of all trees in an area are added together to determine stand density. Stand density affects nearly every decision made regarding thinning, growth and wildlife habitat.

This calculator can be used three ways: to calculate the cross-sectional area of a single tree from its diameter, to convert trees measured on an area into density per hectare or acre and to convert prism or wedge counts into densities within seconds.

What is the trunk cross-section at breast height?

Imagine that you cut the trunk straight through at breast height and look at the flat surface of the cut. This surface is the cross section of the trunk at breast height.

Since the trunk is nearly circular, its cross-section at breast height corresponds to the area of a circle whose diameter is equal to the diameter at breast height (DBH). Since the cross-sectional area of an individual tree is a small number, the values for all trees within a study area are added together and expressed as a quantity per unit area to make it practical. This is usually given in square meters per hectare or square feet per acre.

The breast height diameter is one of the most useful indicators in forestry. It shows how much space each tree occupies on a site, tracks competition between trees and is used to estimate timber volume, growth, crown cover and wildlife habitat.

Formula for calculating trunk cross-section at breast height.

For individual trees, the cross-sectional area of the trunk at breast height can be directly calculated from its diameter at breast height using the formula for the area of a circle.

BA = \pi \left(\frac{DBH}{2}\right)^{2} = \frac{\pi}{4}\, DBH^{2}

The unit of the result is the square of the unit in which the diameter was measured. If the DBH is measured in centimeters, then the unit of cross-sectional area at breast height will be square centimeters; if it is measured in inches, then the unit will be square inches. The following two simplified formulas include conversion factors so that the result is directly given in units commonly used in forestry.

If the diameter is given in centimeters and the result should be in square meters:

BA = 0.00007854 \times DBH^{2}\quad(\text{cm} \rightarrow \text{m}^{2})

If diameter is given in inches and result wanted in square feet:

BA = 0.005454 \times DBH^{2}\quad(\text{in} \rightarrow \text{ft}^{2})

0.005454 is a forestry constant that combines the division by four, the factor of pi and the conversion from square inches to square feet into one number.

Example calculation for a single tree:

Suppose the diameter at breast height (DBH) of a tree is 14 inches. Squaring the DBH gives us 196. Multiplying this by a forestry constant will give us the cross-sectional area of the trunk in square feet.

BA = 0.005454 \times 14^{2} = 1.07\ \text{ft}^{2}

Calculate the cross-sectional area of the stem per acre:

The actual goal is the stock density and the calculation procedure is not complicated. One selects a representative survey area from the stock, measures the diameter (DBH) of each tree in this area, calculates the stem cross-sectional area for each tree, adds these values and then divides them by the area of the survey area.

Assume that a square study area of 30 feet on each side is established, for an area of 900 square feet, and within this area there are six trees with diameters at breast height of 14, 8.5, 11, 13.5, 9, and 11.5 inches respectively. The sum of the cross-sectional areas is approximately 4.28 square feet.

Since the area surveyed is 900 square feet which is equal to an area of 0.0207 acres, we can divide the sum of the cross-sectional areas by the survey area in order to obtain a value that is representative of one acre overall:

\frac{4.28\ \text{ft}^{2}}{0.0207\ \text{acre}} \approx 207\ \text{ft}^{2}/\text{acre}

If you round each tree's basal area to the nearest whole number first and then add up the results, the total will be slightly lower. So for the same area surveyed the result could be quoted as about 206. This calculator maintains full accuracy throughout the process. If you switch to the total area mode and enter six diameters, this calculator will give the same density but also show the number of trees and mean quadratic diameter.

Square mean diameter

When measuring a survey area the tool also calculates the quadratic mean diameter (QMD). This is the diameter of a tree whose stem cross-sectional area equals the average stem cross-sectional area of all trees in the area. It is a natural average value which is used in combination with the stem cross-sectional area.

QMD = \sqrt{\frac{\sum DBH^{2}}{n}}

The quadratic mean diameter is slightly larger than the normal average diameter because the diameters are squared and then averaged. The greater the variation in tree sizes, the greater the difference between the two values. For a plot with six trees, the QMD would be about 11.4 inches.

Prism method or angle measuring device

Measuring each individual tree would be comprehensive but time consuming. In practice, foresters usually use wedge-shaped prisms or angle gauges. This method is called point sampling and was developed by Walter Bitterlich.

A point is selected and the prism is allowed to rotate a full turn while looking at the surrounding area. All tree trunks with a diameter larger than that of the prism are counted. When this number is multiplied by the instrument's basal area factor, it gives directly the basal area per acre or hectare without recording the diameter of each tree.

BA\ \text{per acre} = \text{BAF} \times \text{average trees counted}

If the unit is square feet per acre, a factor of 10 is common, while if using square meters per hectare factors of 2 or 4 are often used. Within the stand several points are selected and results averaged. In prism mode a calculation tool does the calculations.

Applications of the breast height diameter area

The breast height diameter area is the basis for stand density management rules. These rules allow foresters to determine if a stand is under, optimum or over stocked and whether thinning is required.

In the simulation of growth and development of individual trees, DBH area can represent degree of competition. It can also be converted into total volume or biomass of stand. Wildlife biologists also use this indicator since crown cover and structure of a stand are closely related to DBH area.

Symbols used:

The symbols used in the above formula are summarized in the table below.

Symbol	Meaning	Example
DBH	Diameter at breast height	14 in
BA	Basal area of a tree	1.07 sq ft
n	Number of trees on the plot	6
QMD	Quadratic mean diameter	11.4 in
BAF	Basal area factor of a prism	10

This calculation tool is a general educational and planning tool. When conducting resource surveys or management decisions please refer to the applicable on-site tree measurement guidelines as well as regional requirements.

Frequently asked questions

How do you calculate the cross-section of a single tree at breast height?: The cross-sectional area at breast height is the area of a circle calculated from the diameter at breast height. It is therefore determined by multiplying π by the square of the value obtained when the diameter (DBH) is divided by 2. If the diameter is given in inches and the result is to be expressed in square feet, then the simplified formula BA = 0.005454 × (DBH)² can be used. If the diameter is given in centimeters and the result is to be expressed in square meters, then the formula BA = 0.00007854 × (DBH)² can be used. The cross-sectional area of a tree with a diameter of 14 inches would be approximately 1.07 square feet.
How to Calculate Tree Cross-Sectional Area per Acre?: Measure the diameter at breast height (DBH) of each tree within a representative study area, calculate the corresponding cross-sectional area for each tree, add up those values and divide by the area of the study site in acres. For example, if six trees have a total cross-sectional area of 4.28 square feet on a 900-square-foot study plot, then you would divide 4.28 by 0.0207 to get about 207 square feet per acre.
What is the Prism Method or Angle Measurement Method?: This is a quick method that can be used in the field and is called point sampling. You stand at a point and rotate a wedge prism or angle gauge around you to scan the area. Trees that appear wider than the movement of the prism are counted. When this number is multiplied by the cross-sectional-area factor for the instrument, it gives directly the cross-sectional area at breast height per acre or hectare without measuring diameter. By averaging counts from several points, accuracy of estimation can be improved.
Where is height measured at bust level?: In many regions of the world including Australia, Canada and Europe, height is measured at breast height which is approximately 4.3 feet above ground level or 1.3 meters. In the United States, the measurement position for height at breast height is 4.5 feet or 1.37 meters. For slopes, diameter is measured from the uphill side of the tree.
Why is the cross-sectional area at breast height important for forestry?: The cross-sectional area at breast height gives an idea of how much space a tree occupies and allows for the stand density and competition to be measured with one number. Managers use this to assess whether thinning is required in a forest stand, simulate growth, estimate timber volume and biomass, and represent habitat condition. This is because crown cover is closely related to cross-sectional area at breast height.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Basal area
Definition of tree and stand basal area and the DBH formula.
Wikipedia: Diameter at breast height
How and where breast-height diameter is measured.
Wikipedia: Forest inventory
Fixed-area plots and point (prism) sampling in stand assessment.