Common mensuration formulae ©

There is a common misconception that the study of forest mensuration requires the memorisation of large quantities of complex and archaic formulae. Fortunately, this is not so. A few simple formulae need to be remembered and used to derive more complicated formulae if necessary.

The formulae used in mensuration basically relate to dimensions of simple Euclidian shapes - lines, planes, circles, cubes, etc. The dimensions that are usually needed are:

1. Lengths of lines (e.g. the distance between two points)
2. Areas of planes (e.g. the size of an inventory plot or forest compartment)
3. Volumes of solids (e.g. the amount of wood in a log or size of a nesting hollow for a tree dwelling animal).

Unfortunately, nature rarely contains objects that are truly like Euclidean shapes:

Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth nor does lightning travel in a straight line (Mandelbrot)

Fortunately, many parts of trees and other the natural objects of interest to the forest mensuration approximate Euclidian shapes. The job of the mensurationist is to select the Euclidean shape that most closely approximates the natural shape and work out the relevant dimensions using the appropriate formula. The dimensions of the natural shape will approximate the dimensions thus calculated. This introduces a source of error that needs to be considered when using the dimensions calculated.

Length and distance

• The distance (D) between points a and c given the slope distance (S) and angle of slope (Q) is calculated as:
  D = S x Cos(Q).
• The distance (V) between points b and c (b perpendicular to c) given horizontal distance (D) and angle of slope (Q) is calculated as:
  V = D x Tan(Q)
• The distance (V) between points b and c (b perpendicular to c) given slope distance (S) and angle of slope (Q) is calculated as:
  V = S x Sin(Q)
• The circumference of a circle (C) with given diameter (D) is calculated as:
  C = PI*D

Area

The area (A) of a square of side L is calculated as: A = L^2
The area (A) of a rectangle of sides sides L and M is calculated as: A = L x M
The area (A) of a circle of diameter given D is calculated as: A = PI*D^2/4
The area (A) of an ellipse of large diameter D and small diameter d is calculated as: A = PI x D x d / 4
The area (A) of a triangle of sides D, E, F (and without knowing the internal angles) is calculated as: A = SQRT(S x (S-D) x (S-E) x (S-F)) where S = (D + E + F) / 2
The area (A) of a 4-sided shape with none of the internal angles necessarily equal or sides parallel, is calculated as 2 triangles.

Volume

The volume (V) of a cube of side L is calculated as: V = L^3
The volume (V) of a block of sides L, M and H is calculated as: V = L x M x H
The volume (V) of a cylinder of diameter D and height H is calculated as: V = H x PI x D^2 / 4
The volume (V) of a conoid of basal diameter D and height H is calculated as: V = H x PI x D^2 / 12
The volume (V) of a part (or frustum) of a conoidoid of basal diameter D, top diameter d and height H is calculated as: V = H x (D^2 + d x D + d^2 ) x (PI / 12)
The volume of a part (or frustum) of a second degree paraboloid is particularly important in forest mensuration because many logs and sections of trees approximate this shape. Several basic formula are used to measure log volume. Assume that S is the basal sectional area, s is the top basal area, s0.5 is the mid-section sectional area, and H the height or length, then the volume (V) is given by: Smalian's formula: V = H x (S + s) / 2 Huber's formula: V = H x s0.5 Newton's formula- Assuming D is basal diameter, d is top diameter and d0.5 is mid-section diameter: V = H x (D^2 + 4 x d0.5 + d^2) x (PI / 24)