The following two methods are generally adopted to find the areas from the previously plotted plans. The methods are: 1. Graphical Method. 2. Instrumental Method.

**1. Graphical Method:**

In this method, the required data is obtained from measurements of the plan.

**The area may be found:**

(i) By the help of some geometrical figures or

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(ii) By the application of various formulae.

**(i) Area by Geometrical Figures:**

**The area may be found out by any of the following methods: **

**(a) By Division into Triangles:**

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This is the most common method. In this method, the whole figure is divided into a number of triangles. The base and altitude of fig. 4.1 each triangle are scaled and its area is found out by multiplying half the base by altitude.

When the boundaries are irregular, they are replaced by straight lines known as equalising or give and take lines. These lines are so drawn that they exclude (give) as much areas as they include (take) as in fig. 4.1. The figure is thus replaced by a polygon which is then divided into triangles.

**(b) By Division into Squares:**

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It consists in placing over the figure piece of tracing paper ruled out into squares (fig. 4.2) each square representing a definite amount of area. The number of complete squares within the boundary is counted and in addition to the incomplete squares which are more than half are counted as complete and those less than half are neglected. The required area is equal to the total number of squares multiplied by the amount of area which each square presents.

**(ii) Areas by the Application of Formulae:**

Various formulae such as Trapezoidal rule, Simpson’s rule may be applied for finding the areas enclosed between the adjacent survey lines and the curved boundaries. This method is suitable for long narrow strips of ground as those occupied by a railway or a road. A base line is taken through the area and divided into a number of equal parts.

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The ordinates at each of the points of division are drawn and measured. The trapezoidal rule assumes that the boundaries between the extremities of the ordinates are straight, while Simpson’s rule assumes that they are portions of parabolic arcs and is, therefore, sometimes called as the Parabolic rule.

**(a) The Trapezoidal Rule:**

In this method, the area is assumed to be divided into series of trapezoids. This is more accurate than the first two methods.

**This may be stated as follows: **

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**“Add half the sum of the first and last ordinates to the sum of the remaining intermediate ordinates. The total sum thus obtained multiplied by the common distance between the points, gives the required area.” **

**The formula may be derived as under:**

In fig 4.3 , O_{0},O_{1},O_{2}…….O_{n}

= the ordinates taken at equal intervals.

d= the common distance between the ordinates

We know that the area of a trapezoid = half the sum of the parallel side x perpendicular distance between them.

**(b) Simpson’s Rule:**

In this method, the boundaries between the ordinates are assumed to be arcs of a parabola. The results obtained by using Simpson’s rule are more accurate and therefore, it is used only when great accuracy is required.

**It may be stated as follows: **

**“Add the sum of first and the last ordinates to the sum of twice the remaining odd ordinates and four times the even ordinates. The total sum thus obtained multiplied by one third of the common distance, gives the required area.” **

**The formula may be derived as under: **

In fig 4.4, required area = the area (Aab’ cC) enclosed between the base line AC, the parabolic arc ab’c , and the end ordinates Aa and Cc.

Through b’ draw a’ b’ c’ parallel to the chord abc.

Now, the area Aab’cC= area of trapezoid AacC+ area of the segment ab’cba between the parabolic arc ab’c and the chord abc.

**Area of trapezoid AacC: **

**Note: **

(i) As we take two divisions at a time such as 1st and 2nd, 3rd and 4th etc., therefore, this method necessitates an even number of divisions of the area i. e. the total number of ordinates must be odd. If there be an odd number of divisions the area of the last division must be calculated separately and added to the result obtained by applying Simpson’s rule to the remaining divisions.

(ii) For the applications of Simpson’s and Trapezoidal rules, the interval between the successive ordinates must be uniform throughout the length of the base line. If it is not the same, then the base line may be divided into different sections, each having the same interval. The area of these sections should be calculated separately and added together to obtain the required area.

**Example 1:**

**The following perpendicular offsets were taken at 20m intervals from a base line to an irregular boundary line: **

5.9, 12.4, 16.5, 15.3, 18.4, 20.9, 24.2, 21.8 and 19.2 metres.

Calculate the area enclosed between the base line, the irregular boundary line and the first and last offsets by (i) Trapezoidal rule (ii) Simpson’s rule.

**Solution: **

The common distance between the offsets, d = 20 m

**(i) Area by Trapezoidal Rule:**

**(ii) Area by Simpson’s Rule:**

**Example 2:**

**The following offsets were taken from a chain line to a hedge: **

Calculate the area enclosed by the chain line, the hedge and the end offsets by (a) Simpson’s Rule (b) Trapezoidal Rule.

**Solution:**

When the interval between the offsets is not regular through the survey line, the line may be divided into different sections each having a uniform interval. The area of the different sections may be calculated separately and then added together to obtain the required area. In this problem, the survey line is divided into three sections, the first one having 20 m interval, the second one having 40 m interval and the third one having 30 m interval between the offsets.

Let A = the required total area

A_{1} = the area of the first section

A_{2} = the area of the second section.

A_{2} = the area of the third section.

**(i) By Simpson’s Rule: **

**(ii) By Trapezoidal Rule: **

**2. Instrumental Method:**

This method consists in determining the area of a given figure by the use of planimeter. This is a mechanical but the best and most expeditious method, and gives far more accurate results that, those obtained by other methods except by the method, where direct use of field-notes is made.

**There are two types of the planimeter:**

(i) The rolling planimeter, and

(ii) The Amsler polar planimeter, the second type being in common use.

**The Amsler Polar Planimeter (Fig. 4.5): **

It was invented by Prof. J. Amsler Loffon in 1854. It consists of two arms hinged together at the hinge (B) as shown. One of the two arms is known as Pole arm (E) and is of fixed length. It carries a needle point or pole point (F) which is pressed into the paper mounted on a plane table or a drawing board.

The other arm called the tracing arm (A) is of an adjustable length and can slide in a sleeve and clamped to any scale. Its one end carries the tracing point (D) by means of which the outline of the figure to be measured is very carefully followed by moving it always in a clockwise directions.

The recording or measuring wheel (C) is divided into 100 parts of its circumference, the tenths or a part being read on the vernier. The number of complete revolutions made by the roller or the wheel are read on the counting disc or dial.

The counting disc is divided into 10 equal parts and advances one line at every turn of the wheel and performs one revolution at every 10 turns of the wheel. Thus each reading consists of four digits, the units being read on the counting disc, the tenths and hundredths on the measuring wheel, and the thousandth on the vernier.

When the instrument is placed on the paper ready for use, there are three points of contact viz, the tracing point, the pole point and the rim of the measuring wheel. When the tracing point is moved round the outline of the figure, the wheel partly rotates and partly slips and slides.

As the axis of the wheel partly rotates to the tracing arm and the plane of wheel is perpendicular to the tracing arm, the normal component of the motion causes rotation while the axial component of the motion causes slip without affecting the reading. Thus the rotation of the wheel measures the total normal displacement and thereby the area of the figure.

**Planimeter can be used in two ways: **

(i) With the pole point outside the figure, and

(ii) With the pole point inside the figure.

The first method is suitable for measuring small areas while the second is used for large areas. However if the area is large and it is desired to place the pole outside the figure, the figure may be divided into parts, the area of each part being measured separately with the pole point outside the outline, and the results added together to obtain the required area.

**(i) Pole Point Outside the Figure:**

Area of the figure is equal to distance between the tracing point and the hinge point multiplied by the product of the circumference of the measuring wheel and the number of revolutions made by the wheel.

**(ii) Pole Point Inside the Figure:**

In this case the area of the zero circle expressed as a constant and usually given by the maker will have to be added to the above area as given in (i) because the planimeter in this case records only the area of the annular space between the given figure and the zero circle

**Zero Circles:**

It is the circle round the circumference of which, if the tracing point is moved, no rotation of the wheel takes place but actually the wheel only slides on the paper without changing the reading. This happens when the line joining the pole point to the wheel is at right angles to the line joining the tracing point to the wheel. The pole point is the centre of the zero circle and the line joining the pole point to the tracing point is its radius.

**The area of the zero circle may be determined by the following methods: **

**(i) By Using the Planimeter: **

Find the area of a figure of known area with the pole point inside the figure. The area of the zero circle may then be found out by comparing the measured area to the known area of the figure.

**Note: **

If the area of the figure is not known, then first find its area by the planimeter with the pole point outside the figure and then proceed as above.

(ii) By Using the Formulae,

(a) Radius of the zero circle, R

where, L = the length of the tracing arm from the hinge to the tracing point.

L_{1} = the distance from the hinge to the wheel.

L_{2} = the length of the pole arm from the hinge to the pole point.

Plus sign is used when the wheel is placed beyond the hinge and away from tracing point While minus is used when it is placed between the hinge and the tracing point.

.** ^{.}**. Area of the zero circle πR

^{2}= n (L

^{2}± 2LL

_{1}+ L

_{2}

^{2})

(b) Area of zero circle = M × C

Where, M = the multiplier whose value is marked next to the scale division.

C = the constant whose value is either marked on the top of the tracing arm just above the scale division or supplied by the maker on a separate chart.

**Use of the Planimeter:**

**The use of the Planimeter is explained in the following steps: **

(i) Set the index mark on the bevelled edge of the slide to the scale to which the figure has been drawn.

(ii) Move the tracing point around the figure so as to see whether all points on the periphery of the figure can be reached without any difficulty, and then fix the pole point firmly.

(iii) Mark a definite point called dead point on the outline of the figure and set the tracing point exactly on it.

(iv) Read the counting disc and wheel and record it as the initial reading (I.R.)

(v) Move the tracing point in the clockwise direction unless it reaches again at the starting point. Again read the disc and wheel recording as the final reading (F.R.).

(vi) Note the number of times the zero mark of the counting disc passes the fixed index mark in a clockwise or counter-clockwise direction, while the tracing point being moved along the outline of the figure.

**(vii) ****The area of the figure may then by calculated by the formula: **

Area = M (F.R. — I.R. ± ION + C) ……………………… … (Eqn. 4.3)

where, M = the multiplier, the number of units of area per revolution of the measuring wheel whose value is marked on the tracing arm adjacent to the scale division.

F.R. = the final reading

I.R. = the intial reading

N = the number of times the zero mark of the counting disc passes the fixed index mark. Use plus sign if the zero mark passes in a clockwise direction (e.g. 6, 7, 8, 9, 0, 1, 2, etc.) and minus sign if in a counter clock-wise direction (e.g. 3, 2, 1, 0, 9, 8, 7, etc.)

C = the constant to be added only if the pole point is fixed inside the figure and whose value is either marked on the top of the tracing arm just above the scale division or supplied by the maker on a separate chart.

**Precautions:**

**The following precautions should be observed in using the planimeter: **

i. The plan should be kept in a horizontal position and the surface of the paper must be smooth.

ii. As far as possible, the pole point should be fixed outside the figure.

iii. While moving, the tracing point must be kept throughout exactly on the outline of the figure. For this tracing point may be guided by a straight edge or French curve along the outline of the figure.

iv. The area of the figure should be found out twice or thrice with different positions of the pole point and the mean should be taken.

v. The area of the figure should be found out roughly estimation and compared with that obtained by the plan-meter as a check.

**Example 3:**

The following readings were obtained when a figure was traversed clockwise with the anchor point and with the tracing arm set to the natural scale giving the multiplier as outside 100 sq.cm. The zero mark of the counting disc passed the fixed index mark twice in the clockwise direction.

Initial reading = 8.378; final reading = 1.436

Calculate the area of the figure

**Solution: **

From Eqn. 4.3, Area = M (F.R. – I.R. ± 10N + C)

The anchor or pole point being fixed outside the figure, the constant C is not to be added. Since the zero mark of the counting disc passed the fixed index mark twice in the clocks wise direction, the value of N is 2 and is positive.

Area = M (F.R. – I.R. + 10N)

= 100 (1.436 – 8.378 + 10 x 2)

= 100 (13.058) = 1305.8

∴ Area of the figure = 13056.8 Sq.cm (Ans.)

**Example 4:**

Calculate the area of a figure drawn to a scale of 10 metres to 1 cm traversed clockwise with the pole-point inside and with the tracing arm set to the natural scale (metric system) from the following readings:

Initial reading = 4.547

Final reading = 8.932 and C= 23.535

The zero mark of the dial passed the fixed mark once in the reverse direction.

**Solution:**

**Since the scale is 10 meters to 1cm, therefore, 1sq .sm on the paper= 100sq. meters on the ground**