# 10. MEASUREMENT OF AREAS

## 10.1 Introduction

1. One of the main purposes of your topographical survey may be to determine the area of a tract of land where you want to build a fish-farm. From existing topographical maps, you may need to calculate the area of a watershed or of a future reservoir (see Water, Volume 4 in this series).

Note: in land surveying, you should regard land areas as horizontal surfaces, not as the actual area of the ground surface. You always measure horizontal distances.

2. You will often need to know the areas of cross-section profiles to calculate the amount of earthwork you need to do.

 Horizontal area Cross-section area

3. You may determine areas either directly from field measurements, or indirectly, from a plan or map. In the first case, you will find all the measurements of distances and angles you need by surveying, and you will calculate the areas from them. In the second case, you will draw a plan or map first (see Chapter 9). Then you will get the dimensions you need from the scale, and determine the area on that basis.

4. There are several simple methods available for measuring areas. Some of these are graphic methods, where you compare the plan or map of the area you need to measure to a drawn pattern of known unit sizes. Others are geometric methods, where you use simple mathematical formulas to calculate areas of regular geometrical figures, such as triangles, trapeziums*, or areas bounded by an irregular curve.

Note: a trapezium is a four-sided polygon with two parallel sides.

5. The simple methods will be described in detail in the next sections. They are also summarized in Table 13.

 Triangle Trapezium 1 Trapezium 2 Irregular area

TABLE 13
Simple area measurement methods

 Section Method Remarks 10.2 Strips Graphic method giving rough estimate 10.3 Square-grid Graphic method giving good to very good estimates 10.4 Subdivision into regular   geometric figures such as, triangles, trapeziums Geometric method giving good to very good estimates 10.5 Trapezoidal rule Geometric method giving good to very good estimates Suitable for curved boundary

## 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.

Scale: 1: 2.000

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.

Total area = 320 m x 20 m = 6400 m2

## 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.

4. Look at the squares around the edge of the drawing. If more than one-half of any square is within the drawing, count and mark it as a full square. Ignore the rest.
Half or more squares

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.

## 10.4 How to subdivide the area into regular geometrical figures

1. When you need to measure areas directly in the field, divide the tract of land into regular geometrical figures, such as triangles, rectangles or trapeziums. Then take all the necessary measurements, and calculate the areas according to mathematical formulas (see Annex 1). If a plan or map of the area is available, you can draw these geometrical figures on it, and find their dimensions by using the reduction scale.

2. In the first manual in this series, Water for Freshwater Fish Culture, FAO Training Series (4), Section 2.0, you learned how to calculate the area of a pond using this method. In the following steps, you will learn how to apply it under more difficult circumstances.

### 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
 Area = �s(s - a) (s - b) (s - 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)
 Area = (bc sin BAC) � 2

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

 Degree Sine Degree Sine Degree Sine 1 0.0175 31 0.5150 61 0.8746 2 0.0349 32 0.5299 62 0.8829 3 0.0523 33 0.5446 63 0.8910 4 0.0698 34 0.5592 64 0.8988 5 0.0872 35 0.5736 65 0.9063 6 0.1045 36 0.5878 66 0.9135 7 0.1219 37 0.6018 67 0.9205 8 0.1392 38 0.6157 68 0.9272 9 0.1564 39 0.6293 69 0.9336 10 0.1736 40 0.6428 70 0.9397 11 0.1908 41 0.6561 71 0.9455 12 0.2079 42 0.6691 72 0.9511 13 0.2250 43 0.6820 73 0.9563 14 0.2419 44 0.6947 74 0.9613 15 0.2588 45 0.7071 75 0.9659 16 0.2756 46 0.7193 76 0.9703 17 0.2924 47 0.7314 77 0.9744 18 0.3090 48 0.7431 78 0.9781 19 0.3256 49 0.7547 79 0.9816 20 0.3420 50 0.7660 80 0.9848 21 0.3584 51 0.7771 81 0.9877 22 0.3746 52 0.7880 82 0.9903 23 0.3907 53 0.7986 83 0.9925 24 0.4067 54 0.8090 84 0.9945 25 0.4226 55 0.8192 85 0.9962 26 0.4384 56 0.8290 86 0.9976 27 0.4540 57 0.8387 87 0.9986 28 0.4695 58 0.8480 88 0.9994 29 0.4848 59 0.8572 89 0.9998 30 0.5000 60 0.8660

 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. Two triangles 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. Radiation from a central station Radiation from a central station Radiation from a lateral station

 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.

Area = (base x height) � 2

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

 Area = (base x height) � 2

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

 Area = height x (base 1 + base 2) � 2

where:

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

Area = Height x (base 1 + base 2) � 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.

 Lot No1 Height (m) Base(m) (B1+B2) / 2 (m) Area (m2) 1 2 1 TR 5.20 6.50 - 3.25 16.90 2 TP 7.65 5.20 6.20 5.70 43.61 3 TR 6.20 17.10 - 8.55 53.01 4 TR 9.65 4.00 - 2.00 19.30 5 TP 10.50 9.65 14.80 12.22 128.31 6 TP 13.95 14.80 11.80 13.30 185.54 7 TR 11.80 2.80 - 1.40 16.52 Total area 463.19

1TR = right-angled triangle; TP trapezium

### Subdividing land areas without base lines

12. When the shape of the land is more complicated than the ones you have just learned to measure, you will have to use more than one base line, and subdivide the area into triangles, and trapeziums of various shapes. Usually there will be no existing right angle for you to work with and you will have to calculate the area of the trapeziums by taking additional measurements, which will determine their heights along perpendicular lines.

Example

Land tract ABCDEFGHIA along a river is subdivided into five lots 1-5 representing three triangles (1,2,5) and two trapeziums (3 with BE parallel to CD, and 4 with EI parallel to FH). The land boundary forms a closed polygon, which has been surveyed as shown.

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

 s = (a + b + c) � 2 area = �s(s-a)(s-b)(s-c)

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:

 Triangle Length x of sides (m) s (m) (s- x) in m Area (m2) a b c (s-a) (s-b) (s-c) 1 650 860 860 1185 535 325 325 258773.25 2 860 980 840 1340 480 360 500 340258.66 5 660 420 360 720 60 300 360 68305.16 Total area of triangles 667337.07

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

 area = height x (base 1 + base 2) � 2

Example

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

Apply the formula in the following table:

 Lot No. Height (m) Base (m) (B1 + B2) / 2 (m) Area (m2) 1 2 3 560 980 600 790 442400 4 460 840 660 750 345000 Total area of trapeziums 787400

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:

 area = (height x base) � 2

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:

 Lot No. Height (m) Base (m) (B1 + B2) / 2 (m) Area (m2) 1 2 1 600 860 - 430 258000 2 810 840 - 420 340200 3 560 980 600 790 2400 4 460 840 660 750 345000 5 206 660 - 330 67980 Total area of land tract 1453580

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.

2. If part of the land tract is bounded on one side by an irregular curve, such as a road or river, you can find its area by using the trapezoidal rule as explained in this section.

3. Set out straight line AB joining the sides of the tract of land and running as closely as possible to the curved boundary. To determine the irregular area ABCDA, proceed as follows.

4. Measure distance AB and subdivide it into a number of regular intervals, each, for example, 22.5 m long. Mark each of the intervals on AB with ranging poles.

Note: the shorter these intervals are, the more accurate your area estimate will be.

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:

 Area = interval x (ho + hn + 2hi) � 2

where:

ho is the length of the first offset, AD;
hn is the length of the last offset, BC; and
hi is the sum of the lengths of all the intermediate offsets.

Example

Interval = 112.5 m � 5 = 22.5 m
ho = 20 m and hn = 10 m
hi = 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, ho and hn are both equal to zero, and the formula becomes:

 Area = interval x hi

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

Example

Interval = 158 m � 6 = 26.3 m
hi = 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.