10. MEASUREMENT OF AREAS

TABLE 13
Simple area measurement methods

10.2 How to use the strips method for measuring areas

1. Get a piece of transparent paper, such as tracing paper or light-weight square-ruled millimetric paper. Its size will depend on the size of the mapped area you need to measure.

2. On this paper, draw a series of strips, by drawing a series of parallel lines at a regular, fixed interval. Choose this strip width W to represent a certain number of metres. You can follow the scale of the plan or map to do this.  

Example

Scale 1: 2 000; strip width W = 1 cm = 20 m.
Scale 1: 50 000; strip width W = 1 cm = 500 m.

Note: the smaller the strip width, the more accurate your estimate of the land area will be.

3. Place the sheet of transparent paper over the plan or map of the area you need to measure, and attach it securely with drawing pins or transparent tape.

4. For each strip, measure the distance AB in centimetres along a central line between the boundaries of the area shown on the map.

5. Calculate the sum of these distances in centimetres. Then, according to the scale you are using, multiply to find the equivalent distance in the field, in metres.

Example

Scale is 1 :2000 and 1 cm = 20 m.
Sum of distances = 16 cm.
Equivalent ground distance: 16 x 20 m = 320 m.

6. Multiply this sum of real distances (in metres) by the equivalent width of the strip W (in metres) to obtain a rough estimate of the total area in square metres.

Example

Sum of equivalent distances is 320 m.
Strip width (1 cm) is equivalent to 20 m.
Land area: 320 m x 20 m = 6 400 m² or 0.64 ha

Note: 10000 m² = 1 hectare (ha)

7. Repeat this procedure at least once to check on your calculations.

10.3 How to use the square-grid method for measuring areas

1. Get a piece of transparent square-ruled paper, or draw a square grid on transparent tracing paper yourself. To do this, trace a grid made of 2 mm x 2 mm squares inside a 10 cm x 10 cm square, using the example given on the page.

Note: if you use smaller unit squares on the grid, your estimate of the land area will be more accurate; but the minimum size you should use is 1 mm x 1 mm = 1 mm².

2. Place this transparent grid over the drawing of the area you need to measure, and attach it to the drawing securely with thumbtacks or tape. If your grid is smaller than this area, start at one edge of the drawing. Clearly mark the outline of the grid, then move to the next section and proceed in this way over the entire area.

3. Count the number of full squares included in the area you need to measure. To avoid mistakes, mark each square you count with your pencil, making a small dot.

Note: towards the centre of the area, you may be able to count larger squares made, for example, of 10 x 10 = 100 small squares. This will make your work easier.

5. Add these two sums (steps 3 and 4), to obtain the total number T of full squares.

6. Add the sums again at least once to check them.

7. Using the distance scale of the drawing, calculate the equivalent unit area for your grid. This is the equivalent area of one of its small squares.  

Example

• Scale 1:2000 or 1 cm = 20 m or 1 mm = 2 m
• Grid square size is 2 mm x 2 mm
• Equivalent unit area of grid = 4 m x 4 m = 16 m²

8. Multiply the equivalent unit area by the total number T of full squares to obtain a fairly good estimate of the measured area.  

Example

• Total count of full squares T = 256
• Equivalent unit area = 16m²
• Total area = 256 x 16 m² = 4096 m²

Note: when you work with large-scale plans such as cross-sections, you can improve the accuracy of your area estimate by modifying step 5, above. To do this, look at all the squares around the edge of the drawing which are crossed by a drawing line. Then, estimate by sight the decimal part of the whole square that you need to include in the total count (the decimal part is a fraction of the square, expressed as a decimal, such as 0.5, 0.1 and 0.9).

Example

Square A = 0.5; B = 0.1; C = 0.9.

Measuring areas by triangles

3. You can easily calculate the area of any triangle when you know the dimensions of:

• all three sides a, b and c

where s = (a + b + c) � 2;

Example

If a = 35 m; b = 29 m; and c = 45.5 m. Then s = (35 m + 29 m + 45.5 m) � 2 = 54.75 m

Area² = 54.75 m (54.75m - 35 m) (54.75 m - 29 m)(54.75 m - 45.5 m)
= 54.75 m x 19.75 m x 25.75 m x 9.25 m = 257 555 m⁴

Area = �(257 555 m⁴) = 507 m²

• two sides (b, c) and the angle BAC between them (called the included angle)

obtaining sin BAC from Table 14.

Example

If b = 29 m; c = 45.5 m; and angle BAC = 50�.
Then sin BAC = 0.7660 (Table 14)
Area = (29 m x 45.5 m x 0.7660) � 2 = 1010.737 � 2 = 505.3685 m²

  TABLE 14

Sine values of angles

4. Subdivide the tract of land into triangles. For a four-sided area, you can do this in two ways.

• You can join two opposite angles with a straight line BD. Measure the length of BD to find the length of the three sides of each of the two triangles, then calculate their areas (see step 3, above). The sum of the two triangular areas is the total area.
• You can proceed by radiating from central station 0. Measure consecutive angles AOB, BOC, COD and DOA. Then measure distances OA, OB, OC and OD from 0 to each corner of the site and calculate the area of each triangle (see step 3, above). The sum of the four triangular areas is the total area.

5. On a land tract with more than four sides, you can subdivide its area into triangles:

• by radiating from a central station 0 (see step 4, above); or
• by radiating from a lateral station, such as A.

6. Check on your calculations. If you have found the area by using two opposite angles, use the first procedure. If you have proceeded by radiating, use the second.

• Repeat the measurement of the total area by using the other two triangles ABC and ACD, formed by straight line AC.
• Alternatively repeat the measurements of angles and lengths from either the same station or a different one.

Using a base line to subdivide land areas

7. When the shape of the land is polygonal*, you should usually subdivide the total area you need to measure into a series of regular geometrical figures (1-7 in the example) from a common base line AD. You will lay out offsets from the other summits of the polygon* which are perpendicular to this base line to form right triangles 1,3,4 and 7, and trapeziums 2, 5 and 6.

8. When you are choosing a base line, remember that it should:

• be easily accessible along its entire length;
• provide good sights to most of the summits of the polygon;
• be laid out along the longest side of the land area to keep the offsets as short as possible;
• join two polygon summits.

9. Calculate the area of each right-angled triangle*, using the formula:

10. Calculate the area of each trapezium, using the formula:

where:

• Base 1 is parallel to base 2;
• Height is the perpendicular distance from base 1 to base 2.

11. Add together all these partial areas to find the total land area. You should use a table to enter alI the basic dimensions for both right triangles (one base) and trapeziums (two bases), as shown in the example.

Example

• Along base line AD, measure from point A cumulative distances to points H, I, J, K, L, and D, as follows:

Base line (in m)

• From these measurements, obtain partial distances for each section AH, HI, IJ, JK, KL and LD as follows:

Base line (in m)

• Measure offsets HG, IB, ... LE from the base line to each polygon summit:
  HG = 11.80 m; lB = 5.20 m; ... LE = 9.65 m
• Enter these data in the following table, and obtain partial areas of each lot 1, 2, 3, 4, 5, 6 and 7; the sum is the total area.

Subdividing land areas without base lines

13. Calculate the areas of triangles 1, 2 and 5, using the lengths of their three sides and the following formulas:  

Example

Take measurements of the sides of the triangles, as necessary.

Apply the formula area = �s(s- a)(s- b)(s-c) in the following table:

14. Calculate the areas of trapeziums 3 and 4, determining their heights and base lengths, and using the following formula:

Example

Measure the heights and bases of the trapeziums, as necessary.

Apply the formula in the following table:

15. Add the total area of the triangles (step 12) to the total area of the trapeziums (step 14) to obtain total land tract area.

Example

Total area of triangles     =   667337 m²
Total area of trapeziums =   787400 m²
Total land tract area       = 1454 737 m²
                                    or  145.47 ha

16. Another way of making the calculations easier is to measure from a plan the height of each triangle along the perpendicular laid out from one angle summit to the opposite side (called the base). Then, to calculate each triangle area as:

Enter all the data in a single table, as explained in step 11, above.  

Example

From a plan, measure heights BJ, BK and LG for triangles 1,2, and 5, respectively.

Enter all the data in the following table:

The total area of the land tract is 145.36 ha, which is slightly different from the previous estimate (see step 15). This was caused by scaling errors when measuring from the plan, which in this case are small enough (0. 11 ha or 0.07 percent) to be permissible.

10.5 How to measure areas bounded by a curve

1. In Volume 4 of this series, Water for Freshwater Fish Culture (see Section 2.0), you learned how to calculate the area of a pond that has one curving side. You can use a similar procedure to determine the area of a land tract bounded by a regular curve, by trying to balance the partial areas.

5. At each of these marked points, set out a perpendicular line joining AB to the curved boundary. Measure each of these offsets.

6. Calculate area ABCDA using the following formula:

where:

h_o is the length of the first offset, AD;
h_n is the length of the last offset, BC; and
h_i is the sum of the lengths of all the intermediate offsets.

Example

Interval = 112.5 m � 5 = 22.5 m
h_o = 20 m and h_n = 10 m
h_i = 27 m + 6 m + 14 m + 32 m = 79 m
Area ABCDA = 22.5 m x (20 m + 10 m + 158 m) � 2 = (22.5 m x 188 m) � 2 = 2115 m²

Note: remember that you must still calculate the area of AXYBA and add it to the area of ABCDA to get the total area DAXYBCD.

7. If you can lay out line AB so that it touches the two ends of the curved boundary, your calculations will be much simpler. In this case, h_o and h_n are both equal to zero, and the formula becomes:

where hi is the sum of the lengths of all the intermediate offsets.

Example

Interval = 158 m � 6 = 26.3 m
h_i = 25 m + 27 m + 2 m + 23 m + 24 m = 101 m
Area= 26.3 m x 101 m = 2 656.3 m²

Note: remember that you must still calculate the area of AXYBA and add it to the area of the curved section to get the total area.