# 5. CALCULATING SURFACE AREAS OF IRREGULAR SHAPED FIELDS

5.1 Example 1
5.2 Example 2

A common problem for a surveyor is the calculation of the surface area of a farmer's field. The fields are often irregular which makes direct calculation of their areas difficult. In such case fields are divided into a number of regular areas (triangles, rectangles, etc.), of which the surfaces can be calculated with simple formulas. All areas are calculated separately and the sum of these areas gives the total area of the field.

## 5.1 Example 1

Figure 29 shows a field with an irregular shape of which the surface area must be determined.

Step 1

Make a rough sketch of the field (see Fig. 29a) indicating the corners of the field (A, B, C, D and E) and the field borders (straight lines). In addition some major landmark! are indicated (roads, ditches, houses, trees, etc.) that may help to locate the field.

Fig. 29a A rough sketch of the field

Step 2

Divide the field, as indicated on the sketch, into areas with regular shapes. In this example, the field can be divided into 3 triangles ABC (base AC and height BB,), AEC (base AC and height EE1) and CDE (base EC and height DD1) (see Fig. 29b).

Fig. 29b Division of the field into areas with regular shapes

Step 3

Mark, on the field, the corners A, B, C, D and E with pegs.

Step 4

Set out ranging poles on lines AC (base of triangles ABC and AEC) and EC (base of triangle EDC) (see Fig. 29c) and measure the distances of AC and EC.

Step 5

Set out line BB (height of triangle ABC) perpendicular to the base line AC (see Fig. 29d) using one of the methods described in Chapter 4. Measure the distance BB,

Step 6

In the same way, the height EE, of triangle AEC and the height DD, of triangle CDE are set out and measured (see Fig. 29e)

Step 7

The base and the height of the three triangles have been measured. The final calculation can be done as follows:

Measured

Triangle ABC: base = AC = 130 m
height = BB1 = 55 m
Triangle ACE: base = AC = 130 m
height = EE1 = 37 m
Triangle CDE: base = EC = 56 m
height = DD1 = 55 m

Area = 0,5 x base x height
= 0.5 x 130 m x 55 m = 3 575 m2

Area = 0.5 x 130 m x 37 m = 2 405 m
Area = 0.5 m x 56 m x 55 m= 1 540 m²

Field ABCDE:

Area of triangle ABC = 3 575 m2
Area of triangle ACE = 2 405 m2
Area of triangle CDE = 1 540 m2

Total Area = 3 575 m2 + 2 405 m2 + 1 540 m2
= 7 520 m- = 0.752 ha

## 5.2 Example 2

The surface area of the field shown in Fig. 30 has to be determined at a time that the field is covered by a tall crop (e.g. maize or sugarcane).

The field can be divided into two triangles ABD and BCD (see Fig. 31a). Unfortunately, because of the tall crop, setting out and measurement of the base BD and the two heights AA1 and CC1 is impossible.

Fig. 31a Division of the field in two triangles

In this case, the area of triangle ABD can be calculated using AD as the base and BB1 as the corresponding height. BB1 can be set out and measured outside the cropped area. In the same way, triangle BCD can be calculated using base BC and the corresponding height DD1 (see Fig. 31b).

Fig. 31b Determination of the areas of the two triangles

The procedure to follow on the field is:

Step 1

Mark the 4 corners (A, B, C and D) with ranging poles.

Step 2

Line AD is set out with ranging poles and extended behind A. Line BC is also set out and extended behind C (see Fig. 32a). Measure the distances AD (base of triangle ADB) and BC (base of triangle BCD).

Step 3

Set out line BB1 (height of triangle ABD) perpendicular to the extended base line AD using one of the methods described in Chapter 4. In the same way, line DD1 (height of triangle BCD) is set out perpendicular to the extended base line BC (See Fig. 32b) Measure the distance BB1 and DD1.

Step 4

The base and height of both triangles have been measured. The final calculations can be done as follows:

Measured

Triangle ABD: base = AD = 90 m
height = BB1 - 37 m

Triangle BCD: base = BC = 70 m
height = DD1 - 50 m