1. 1
Chapter 9 Areas and volumes
Traverse Area Computations
Trapezoidal rule
The trapezoidal rule is a numerical integration method used to approximate the area under a
curve with a number of trapezoidal areas. Consider an irregular area shown in Figure below.
The area is sub-divided into 5 strips with the offset specified at every 10 m interval. The area
is expressed by the following equation:
Area = (x/2) [O1 + 2O2 + 2O3 + … + On]
Where, x = distance or width of each strip
O1,2,…n = first, second, … last offset readings
By using the trapezoidal equation, the area can be estimated as follows:
Area = (x/2) [O1 + 2O2 + 2O3 + … + On]
= (10/2) [10.2 + 2 (14.5) + 2 (14.3) + 2 (10.6) + 2 (13.1) + 9.8]
= 625 m2

2. 2
Simpson’s rule
Similar to the trapezoidal rule, Simpson’s rule is another form of numerical integration
method. This method is often used when more accurate area or volume is required
particularly for construction projects that use costly material. It assumes that the irregular
boundary is composed of a series of parabolic arcs.
Consider the example shown in the Figure below. It is the same example as in previous
Figure. However, the irregular boundaries in this example are no longer assumed to be
straight lines, but a series of parabolic arcs. Therefore, the area estimates with this approach
is more accurate.
The area estimated by Simpson’s rule is expressed by the following equation:
Area = (x/3) [O1 + 2O + 4E + On]
Where, x = distance or width of each strip
O1 = first offset reading
On = last offset reading
O = odd offset readings
E = even offset readings
Let us resolve the above example by using Simpson’s rule:
Area = (x/3) [O1 + 2O + 4E + On]
= (10/3) [10.2 + 2(14.3 + 13.1) + 4(14.5 + 10.6) + 9.8]
= 584 m2

3. 3
Area inside a traverse

4. 4

5. 5

6. 6
Splitting a traverse into two areas

7. 7

8. 8

9. 9
Volumes from a rectangular grid of spot heights

10. 10

11. 11