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.

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

**Fig. 29 A field of irregular shape**

The procedure to follow is:

__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 EE_{1}) and CDE (base EC and height DD_{1}) (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.

**Fig. 29c Mark the corners with pegs and set out ranging poles**

__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,

**Fig. 29d Set out line BB perpendicular to AC**

__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)

**Fig. 29e Set out line DD _{1 }perpendicular to EC and line EE1 perpendicular to AC**

__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 = BB_{1} = 55 m

Triangle ACE: base = AC ^{=} 130 m

height = EE_{1 }= 37 m

Triangle CDE: base = EC *=* 56 m

height = DD_{1 }= 55 m

__Answer__

Area = 0,5 x base x height

= 0.5 x 130 m x 55 m = 3 575 m^{2}

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 m^{2 }

Area of triangle ACE = 2 405 m^{2}

Area of triangle CDE = 1 540 m^{2}

Total Area = 3 575 m^{2} + 2 405 m^{2} + 1 540 m^{2 }

= 7 520 m- *=* 0.752 ha

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).

**Fig. 30 A field covered by a tall crop**

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 AA_{1 }and CC_{1 }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 BB_{1 }as the corresponding height. BB_{1} 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 DD_{1} (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).

**Fig. 32a Measurement of the bases of the two triangles**

__Step 3__

Set out line BB_{1 } (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 DD_{1 } (height of triangle BCD) is set out perpendicular to the extended base line BC (See Fig. 32b) Measure the distance BB_{1 }and DD_{1}.

**Fig. 32b Measurement of the heights of the two triangles**

__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 *=* BB_{1} - 37 m

Triangle BCD: base = BC = 70 m

height = DD_{1} - 50 m

__Answer__

Area = 0.5 x base x height

= 0.5 x 90 m x 37 m = 1 665 m^{2}

Area = 0.5 x 70 m x 50 m = 1 750 m^{2 }

__Field ABDC:__

Area triangle ABD *=* 1 665 m²

Area triangle BCD = 1 750 m^{2}

Total Area = 1 665 m^{2} + 1 750 m^{2} = 3 415 m^{2 }

= 0.3415 ha ^{=} approx. 0.34 ha