A Ruler Palette for Determining Areas from a Plan
Abstract Generated abstract
This paper proposes a simple ruler-palette for determining areas on plans, intended as an alternative to square and parallel palettes or a planimeter, especially where portability and ease of use are important. The method approximates a contour by summing circular sectors formed by successive radius vectors from a chosen center, with a 10 degree working angle and graduations that allow direct reading of sector areas. The author derives estimates for methodological and reading errors, gives procedures for ordinary, elongated, and complex contours, and recommends center placement to keep boundary intersections sufficiently oblique. Experimental comparisons with a planimeter and a parallel palette indicate that the instrument can meet technical accuracy requirements, with somewhat longer measurement times than a planimeter but simpler construction and practical advantages in field or instructional use.
Full Text
Ruler-Palette for Determining Areas from a Plan
G. F. Lysov
(Presented by the scientific seminar of the departments of mine surveying and geodesy)
Depending on the size and configuration of the plots, the determination of areas from a plan is carried out either by a graphical method, or by means of a planimeter or palettes of various kinds. The most convenient for measuring small areas (up to \(10\ \text{cm}^2\) on the plan) are square and parallel palettes. However, as the measured area increases, the time spent determining the area with such palettes increases considerably. In addition, counting the number of whole squares (when using a square palette) or the sum of the lengths of all middle lines (when using a parallel palette) is often accompanied by errors.
In such cases it is more expedient to use a very simple ruler-palette (Fig. 1), constructed by the author on the basis of the following considerations.
Let us divide the measured contour \(ABCD\) (Fig. 2) into \(n\) parts by radius-vectors \(AO\), \(BO\), etc. (shown by dashed lines in Fig. 2), drawn from an arbitrarily chosen center \(O\) inside the contour at a small angle \(\alpha\) to one another. In this case, the area of each curvilinear sector (for example, sector \(ABO\)) \(S_{ABO}\), formed by two adjacent radii \(AO\), \(BO\) and the curvilinear part of the contour boundary \(AB\), will be almost equal to the area of the circular sector \(MFO\) \((S_{MFO})\), with radius equal to \(ON'\) \((R_1)\), drawn along the bisector of the angle \(\alpha\) of sector \(MFO\).
Indeed,
\[ S_{ABO}=S_{MFO}+(S_{FBN}-S_{AMN})+(S_{AB}-S_{FM}), \tag{1} \]
where \(S_{FBN}\), \(S_{AMN}\) are the areas of triangles \(FBN\) and \(AMN\),
\[
S_{AB},\ S_{FM}
\]
are the areas of the segments \(AB\) and \(FM\).
The magnitudes of the areas of the segments \(AB\) and \(FM\) entering into formula (1) are very small. Thus, for example, the area of the segment \(FM\), when \(\alpha = 10^\circ\), amounts to only \(0.5\%\) of the area of the circular sector \(MFO\). With a smooth configuration of the contour boundary, the area of segment \(AB\) will be of the same order of magnitude, and the value of the difference \((S_{AB}-S_{FM})\) is close to zero. Consequently, formula (1) may be written in the following form:
\[ S_{ABO}=S_{MFO}+(S_{FBN}-S_{AMN}). \tag{2} \]
Fig. 1.
It is also clear from Fig. 2 that, for a small value of the angle \(\alpha\),
\[ \angle BFN = 90^\circ + \frac{\alpha}{2} \simeq \angle AMN = 90^\circ - \frac{\alpha}{2} \simeq 90^\circ, \]
and \(\angle BNF = \angle MNA\) (as vertical angles formed by the segments \(AB\) and \(FM\)). It follows from this that, for a sufficiently small value of \(\alpha\), the small triangles \(FBN\) and \(AMN\) may be regarded as similar right triangles with equal legs \(FN\) and \(NM\), i.e., equal in area. Below we shall discuss in more detail the error in determining the area of the curvilinear sector \(S_{ABO}\) and of the entire contour due to the very small inequality of the areas of the triangles \(FBN\) and \(AMN\).
Fig. 2.
Thus, formula (2) can be simplified, i.e.,
\[ S_{ABO} = S_{MFO}, \tag{3} \]
or
\[ S_{ABO} = \frac{\pi R_1^2 \alpha^\circ}{360}\ \text{cm}^2, \tag{4} \]
where \(R_1\) is the radius of the circular sector, equal to \(ON'\) in cm,
\(\alpha^\circ\) is the magnitude of the small angle \(\alpha\) in degrees.
The area of the entire figure \(ABCD\) can therefore be determined by the formula
\[ S_{ABCD}=\sum_{1}^{n} kR^2 \ \mathrm{cm}^2, \tag{5} \]
\[ k=\frac{\pi\alpha^\circ}{360}, \tag{6} \]
where \(\sum_{1}^{n} kR^2\) is the sum of the areas of circular sectors with radii \(R_1, R_2, R_3,\ldots,R_n\), drawn at an angle \(\alpha\) to one another.
The ruler-palette (Fig. 1) is intended for determining areas on a plan by formula (5). The instrument is a circular sector 1 (with a radius of 210 mm), which may be made of plastic, cardboard, wood, or similar material. The sharp end of the ruler is made, for the purpose of providing sufficient strength, as shown in Fig. 1, and terminates in a circle 2 of diameter 10 mm with a round hole 3 of diameter 0.3–0.4 mm. The angle \(\gamma\) between the beveled sides of the ruler 4, 5 is taken equal to \(10^\circ\), since in this case the error in measuring the area (as we shall see below) will be no more than 2%.
In order to speed up the determination of area, the working side 4 of the ruler is graduated in square centimeters in such a way that the reading along this side is equal to the area of a circular sector with an arc of \(10^\circ\) and with a radius equal to the distance from the center 3 to the corresponding mark on the working side of the ruler. The radii (distances from the center 3 to the corresponding marks on the working side of the ruler) may be computed from formula (4) for \(\alpha\) equal to \(10^\circ\), i.e.,
\[ R=\sqrt{\frac{S}{0.0873}}, \tag{7} \]
where \(S\) is the value of the corresponding mark on the working side 4.
Side 5 is graduated in mm and may be used when measuring linear quantities on a plan.
In order to make it possible to use the instrument when measuring areas with very complex contours, strokes 6 are provided on the rounded end of the ruler, made at intervals of \(1^\circ\) along the arc.
The ruler-palette of the size described is intended for determining the areas of contours that can be inscribed in a circle of diameter 400 mm. Larger areas may be measured in parts or with a ruler of size 300–400 mm.
The area of a contour is determined with the aid of the proposed ruler in the following order. Having placed the plan on a level surface, the ruler should be fastened to the plan with a needle or pin through hole 3, positioning center 3 approximately at the center of the contour being measured and so that side 4, in any position of the ruler, intersects the boundary of the contour at an angle of not less than \(30^\circ\). Having marked with a pencil on the plan the point of intersection of side 4 with the boundary of the contour, one takes the reading along this side \((kR_1^2)\) in \(\mathrm{cm}^2\). Then, rotating the ruler in one direction about center 3, it is turned through an angle \(\alpha\) equal to \(10^\circ\), using the mark made earlier on the plan, and the reading is taken along the same side \((kR_2^2)\), again marking the point of intersection of side 4 with the boundary of the contour. Continuing the rotation of the ruler in the same direction and in the same order, one successively determines the values \(kR_3^2, kR_4^2,\ldots,kR_{36}^2\). In doing so, it is convenient to use an abacus and, immediately, without writ-
adding together the found values of \(kR^2\). The total value of the sum thus calculated \(\left(\sum\limits_1^{36} kR^2\right)\) is, in accordance with formula (5), the area of the contour being measured.
When measuring the area of an elongated contour, it is more convenient to use a somewhat different method. In this case one should draw, to the extreme points of the contour \(ABCD\), \(A\) and \(B\) (Fig. 3), the tangents \(AO\) and \(BO\), intersecting at point \(O\) at an angle that is a multiple of \(10^\circ\) (for example, \(90^\circ\)). This construction can be carried out very simply with the aid of a right triangle.
Fig. 3.
Next, having fastened the ruler with a pin through hole 3 at point \(O\), it is necessary to lay off from the tangent \(AO\), using marks 6, an angle of \(5^\circ\), and, after turning the ruler through this angle, as shown in Fig. 3, to take the readings along side 4, \(kR_1^2\) and \(kR_2^2\), and mark point \(C\) on the plan. Continuing the rotation of the ruler in the same direction, one successively takes the readings \(kR_3^2, kR_4^2, \ldots, kR_{n-1}^2, kR_n^2\). Here it is also convenient to use a calculating machine and immediately, without writing down, successively add the quantities \((kR_1^2 - kR_2^2), \ldots, (kR_{n-1}^2 - kR_n^2)\). The resulting total sum
\[ \sum_1^n (kR_{n-1}^2 - kR_n^2) \]
will be the area of the elongated contour.
In the case when the area of a complex contour is measured and side 4 intersects the boundary of the contour \(n\) times, one should take \(n\) readings \(kR^2\) at one position of the ruler; moreover, the nearest (first) reading \(kR^2\) from center 3 and all subsequent odd readings \(kR^2\) (third, fifth, etc.) should be regarded as positive and added to the sum \(kR^2\), while all even readings \(kR^2\) (second, fourth, etc.) should be regarded as negative and subtracted from the sum \(kR^2\).
When measuring the areas of contours of very complex configuration, using marks 6, one may successively rotate the ruler by one, two, or five degrees (depending on the complexity of the configuration of the entire contour or of its part) and take readings \(kR^2\) in the described
order. In this case the resulting sum \(kR^2\) must be reduced correspondingly by 10.5 or 2 times.
For checking and in order to increase accuracy, the area should be determined twice, with different positions of center 3 inside the contour.
The root-mean-square error in determining the area of a contour with the ruler may be computed by the formula
\[ m_S=\pm\sqrt{m_c^2+m_u^2}, \tag{8} \]
where \(m_c\) is the error in determining the area due to the inequality of the triangles \(FBN\) and \(AMN\) (Fig. 2)—the error of the method;
\(m_u\) is the error in determining the area due to errors in reading off the quantities \(kR^2\) on the ruler—the measurement error.
The error of the method \(m_c\) may be determined in the following way. Denoting the angle between the radius \(R_1\) and the straight line \(AB\) (Fig. 2) by \(\beta\) and taking into account that
\[ FN=NM=\frac{\pi R_1\alpha^\circ}{360}, \]
we find the areas of the triangles \(FBN\) and \(AMN\):
\[ S_{FBN}= \frac{k^2R_1^2\cos\beta\cdot\cos\frac{\alpha}{2}} {2\sin\left(\beta-\frac{\alpha}{2}\right)}, \tag{9} \]
\[ S_{AMN}= \frac{k^2R_1^2\cos\beta\cdot\cos\frac{\alpha}{2}} {2\sin\left(\beta+\frac{\alpha}{2}\right)}. \tag{10} \]
In accordance with formula (2), the error in determining the area of the curvilinear sector \(ABO\) \((\Delta S_{ABO})\) due to the inequality of the areas of the triangles \(FBN\) and \(AMN\) is determined from the expressions
\[ \Delta S_{ABO} = \frac{1}{2}k^2R_1^2\cos\frac{\alpha}{2} \left( \frac{\cos\beta}{\sin\left(\beta-\frac{\alpha}{2}\right)} - \frac{\cos\beta}{\sin\left(\beta+\frac{\alpha}{2}\right)} \right), \tag{11} \]
\[ \Delta S_{ABO\%} = 50k\cos\frac{\alpha}{2} \left( \frac{\cos\beta}{\sin\left(\beta-\frac{\alpha}{2}\right)} - \frac{\cos\beta}{\sin\left(\beta+\frac{\alpha}{2}\right)} \right). \tag{12} \]
Examining formula (12), one may conclude that \(\Delta S_{ABO\%}\) depends only on the magnitudes of the angles \(\alpha\) and \(\beta\). Taking \(\alpha=10^\circ\), we determine the value of \(\beta\) for which \(\Delta S_{ABO\%}\) will not exceed 3%; to do this we solve equation (12), substituting into it the value from formula (6), and find that \(\cos\beta=0.890\), i.e. \(\beta=27^\circ\). This means that the position of the ruler center 3 must be chosen so that side 4, for any position of the ruler, intersects the boundary of the contour at an angle of not less than \(30^\circ\).
Denoting
\[ \left( \frac{\cos\beta}{\sin\left(\beta-\frac{\alpha}{2}\right)} - \frac{\cos\beta}{\sin\left(\beta+\frac{\alpha}{2}\right)} \right)=\mu, \tag{13} \]
We shall next find the mean error in determining the area of the entire contour, \(m_c\), using formulas (5), (11), (12)
\[ m_c=\pm \frac{1}{2}\kappa^2 \cos \frac{\alpha}{2}\sqrt{\sum R^4\mu^2}, \tag{14} \]
\[ m_{c\%}=\pm 50\kappa \cos \frac{\alpha}{2}\frac{\sqrt{\sum R^4\mu^2}}{\sum R^2}. \tag{15} \]
Taking \(\alpha=10^\circ,\ \beta_{\mathrm{av}}=45^\circ\), we find
\[ m_c=\pm 6.7\cdot 10^{-4}\sqrt{\sum_{1}^{36}R^4}, \tag{16} \]
\[ m_{c\%}=\pm 0.77\,\frac{\sqrt{\sum_{1}^{36}R^4}}{\sum_{1}^{36}R^2}. \tag{17} \]
It follows from formulas (16), (17) that when \(R_1\approx R_2\approx \ldots \approx R_{36}\), \(m_c\) and \(m_{c\%}\) will be minimal. For example, \(m_{c\%}\) in this case will be equal to
\[ \frac{0.77}{\sqrt{36}}=0.13\%. \]
Thus, choosing the position of center 3 approximately in the middle of the contour reduces the error in determining the area.
For any values \(R_1, R_2,\ldots R_{36}\), the quantity
\[ \frac{\sqrt{\sum_{1}^{36}R^4}}{\sum_{1}^{36}R^2} \]
from formula (17) will be less than unity and, consequently, \(m_{c\%}\) cannot be more than \(0.77\%\). To find the magnitude of the mean error of the area \(m_s\), in the subsequent calculations we shall take the mean value of \(m_{c\%}\) from the possible maximum and minimum values of this error, i.e., \(0.45\%\).
Let us now find the value of \(m_u\). Since the value \(\kappa R^2\) is determined directly on the ruler, the value \(m_u\) may be calculated from the formulas
\[ m_u=\pm \sqrt{\sum_{1}^{36}m_{\kappa R^2}^{\,2}}, \tag{18} \]
\[ m_{u\%}=\pm \frac{100\sqrt{\sum_{1}^{36}m_{\kappa R^2}^{\,2}}}{\sum \kappa R^2}, \tag{19} \]
where \(m_{\kappa R_1^2},\ldots m_{\kappa R_{36}^2}\) are the mean errors of taking readings on the ruler.
Let us denote the variable quantity
\[ \frac{m_{\kappa R_i^2}}{\kappa R_i^2}=\lambda_i. \]
In this case, for the various intervals of the ruler, formulas (18) and (19) take the form
\[ m_u=\pm \kappa \lambda_{\mathrm{av}}\sqrt{\sum_{1}^{36}R^4}, \tag{20} \]
\[ m_{u\%}=\pm \frac{100\lambda_{\mathrm{av}}\sqrt{\sum_{1}^{36}R^4}}{\sum_{1}^{36}R^2}. \tag{21} \]
From formula (21) it follows that \(m_{u\%}\) also depends on the value
\[ \frac{\sqrt{\sum_{1}^{36} R^{4}}}{\sum_{1}^{36} R^{2}} . \]
As has already been noted, for \(R_1 \simeq R_2 \simeq \ldots \simeq R_{36}\) the value
\[ \frac{\sqrt{\sum_{1}^{36} R^{4}}}{\sum_{1}^{36} R^{2}} \]
is equal to 0.167. For any values \(R_1, R_2,\ldots,R_{36}\),
\[ \frac{\sqrt{\sum_{1}^{36} R^{4}}}{\sum_{1}^{36} R^{2}} \]
will be less than unity. Taking for further calculations the mean of the possible minimum and maximum values of the quantity
\[ \frac{\sqrt{\sum_{1}^{36} R^{4}}}{\sum_{1}^{36} R^{2}}, \]
i.e., 0.58, we present formula (21) in the simpler form:
\[ m_{u\%}=\pm 58\lambda_{\mathrm{av}} . \tag{22} \]
We shall summarize the calculation of the quantities \(m_u\) for the various intervals of the ruler in Table 1 below.
Table 1
| No. | Interval on the ruler, cm² | \(\lambda_{\mathrm{av}}\) in the interval | \(m_{u\%}\) | Interval \(S\), cm² |
|---|---|---|---|---|
| 1 | 0.05--0.1 | 0.013 | 0.77 | 1.8--3.6 |
| 2 | 0.1--0.4 | 0.009 | 0.54 | 3.6--14.0 |
| 3 | 0.4--2.0 | 0.0047 | 0.30 | 14.0--72.0 |
| 4 | 2.0--9.0 | 0.0054 | 0.10 | 72.0--320.0 |
| 5 | 9.0--20.0 | 0.0048 | 0.10 | 320.0--720.0 |
| 6 | 20.0--35.0 | 0.0018 | 0.10 | 720.0--1200.0 |
The calculation of the total mean square error in determining the area of a contour with the proposed ruler-palette by formula (8) is presented in Table 2.
Table 2
| No. | Interval \(S\), cm² | \(m_{c\%}\) | \(m_{u\%}\) | \(m_{S\%}\) | \(m_{S\%}\), limiting |
|---|---|---|---|---|---|
| 1 | 1.8--3.6 | 0.45 | 0.77 | 0.9 | 2.7 |
| 2 | 3.6--14.0 | 0.45 | 0.54 | 0.7 | 2.1 |
| 3 | 14.0--72.0 | 0.45 | 0.30 | 0.5 | 1.5 |
| 4 | 72.0--320.0 | 0.45 | 0.10 | 0.45 | 1.3 |
| 5 | 320.0--720.0 | 0.45 | 0.10 | 0.45 | 1.3 |
| 6 | 720.0--1200.0 | 0.45 | 0.10 | 0.45 | 1.3 |
The theoretical calculations obtained are confirmed by experimental investigations carried out according to the following program:
- Measurements were made of the area of a number of specially selected contours (Fig. 4) by means of a planimeter, a parallel palette (with an interval \(h = 2\) mm), and a ruler-palette specially made from drawing paper. The area of each figure was measured 10 times by each of the indicated methods. The dimensions of the figures were selected so that it would be possible to determine the error of area measurement within the ranges \(0\)—\(10\ \mathrm{cm}^2\) and \(10\)—\(100\ \mathrm{cm}^2\). In form, the figures are represented by contours formed by curved and straight lines. At the same time, the time spent on each measurement was determined.
Fig. 4.
- From the results of the actual measurements, the mean errors of one measurement by each of the indicated methods were calculated, and the average expenditure of time for one determination of area was determined for each method.
The results of this work are presented in Tables 3, 4, 5, 6, 7, 8 and in the summary Table 9.
The theoretical and experimental investigations presented of the new method for determining areas from a plan make it possible to draw the following conclusions:
-
The error in determining area on a plan with the proposed instrument meets the requirements imposed on the measurement of areas for technical purposes.
-
The duration of one determination of an area of less than \(10\ \mathrm{cm}^2\) with the ruler-palette is less than the duration of determining area with known palettes.
Table 3
Determination of the area of figure 1
| No. of meas. | Planimeter: area \(S,\ \mathrm{cm}^2\) | Planimeter: \(\varepsilon_s\) | Planimeter: \(\varepsilon_s\varepsilon_s\) | Planimeter: \(t,\ \mathrm{min}\) | Parallel palette: area \(S,\ \mathrm{cm}^2\) | Parallel palette: \(\varepsilon_s\) | Parallel palette: \(\varepsilon_s\varepsilon_s\) | Parallel palette: \(t,\ \mathrm{min}\) | Ruler-palette: area \(S,\ \mathrm{cm}^2\) | Ruler-palette: \(\varepsilon_s\) | Ruler-palette: \(\varepsilon_s\varepsilon_s\) | Ruler-palette: \(t,\ \mathrm{min}\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 5,07 | 0 | 0 | 2,5 | 5,028 | −2 | 4 | 5,0 | 5,047 | −6 | 36 | 6,0 |
| 2 | 5,07 | 0 | 0 | 2,0 | 4,992 | +34 | 1160 | 4,5 | 5,015 | +26 | 676 | 6,0 |
| 3 | 4,98 | +9 | 81 | 2,0 | 5,036 | −10 | 100 | 4,0 | 5,040 | +1 | 1 | 5,5 |
| 4 | 5,02 | +5 | 25 | 2,0 | 5,048 | −22 | 484 | 4,5 | 5,090 | −49 | 2400 | 5,5 |
| 5 | 5,07 | 0 | 0 | 2,0 | 5,000 | +26 | 676 | 4,5 | 5,105 | −64 | 4100 | 6,0 |
| 6 | 5,12 | −5 | 25 | 2,0 | 5,036 | −10 | 100 | 4,0 | 5,020 | +21 | 441 | 6,0 |
| 7 | 5,17 | −10 | 100 | 2,5 | 5,032 | −26 | 676 | 4,5 | 5,030 | +11 | 121 | 5,0 |
| 8 | 4,98 | +9 | 81 | 2,0 | 5,068 | −42 | 1760 | 4,0 | 5,040 | +1 | 1 | 5,0 |
| 9 | 4,98 | +9 | 81 | 1,5 | 5,012 | +14 | 196 | 5,0 | 4,995 | +46 | 2120 | 5,5 |
| 10 | 5,26 | −19 | 361 | 2,0 | 4,992 | +34 | 1160 | 4,0 | 5,025 | +16 | 256 | 5,5 |
| \(\Sigma\ 754\) | \(\Sigma\ 6216\) | \(\Sigma\ 10152\) |
\[ S_{\mathrm{avg.}} = 5,07 \]
\[ m_s = \pm 0,09\ (1,8\%),\quad t_{\mathrm{avg.}} = 2,0 \]
\[ S_{\mathrm{avg.}} = 5,026 \]
\[ m_s = \pm 0,026\ (0,52\%). \]
\[ t_{\mathrm{avg.}} = 4,4 \]
\[ S_{\mathrm{avg.}} = 5,041 \]
\[ m_s = \pm 0,033\ (0,67\%). \]
\[ t_{\mathrm{avg.}} = 5,6 \]
Table 4
Determination of the area of figure 2
| No. of meas. | Planimeter: area \(S,\ \mathrm{cm}^2\) | Planimeter: \(\varepsilon_s\) | Planimeter: \(\varepsilon_s\varepsilon_s\) | Planimeter: \(t,\ \mathrm{min}\) | Parallel palette: area \(S,\ \mathrm{cm}^2\) | Parallel palette: \(\varepsilon_s\) | Parallel palette: \(\varepsilon_s\varepsilon_s\) | Parallel palette: \(t,\ \mathrm{min}\) | Ruler-palette: area \(S,\ \mathrm{cm}^2\) | Ruler-palette: \(\varepsilon_s\) | Ruler-palette: \(\varepsilon_s\varepsilon_s\) | Ruler-palette: \(t,\ \mathrm{min}\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 5,47 | +12 | 144 | 2,0 | 5,412 | +5 | 25 | 7,0 | 5,400 | +31 | 960 | 6,0 |
| 2 | 5,66 | −7 | 49 | 2,0 | 5,408 | +9 | 81 | 6,0 | 5,457 | −26 | 676 | 6,0 |
| 3 | 5,66 | −7 | 49 | 2,0 | 5,392 | +25 | 625 | 7,0 | 5,395 | +36 | 1300 | 6,0 |
| 4 | 5,66 | −7 | 49 | 2,0 | 5,456 | −39 | 1520 | 6,5 | 5,452 | −21 | 440 | 5,5 |
| 5 | 5,77 | −18 | 324 | 1,5 | 5,412 | +5 | 25 | 6,5 | 5,430 | +1 | 1 | 6,0 |
| 6 | 5,66 | −7 | 49 | 1,5 | 5,360 | +57 | 3240 | 6,0 | 5,435 | −4 | 16 | 5,5 |
| 7 | 5,47 | +12 | 144 | 2,0 | 5,448 | −31 | 960 | 6,0 | 5,472 | −41 | 1680 | 6,0 |
| 8 | 5,57 | +2 | 4 | 2,0 | 5,416 | +1 | 1 | 7,0 | 5,400 | +31 | 960 | 6,0 |
| 9 | 5,47 | +12 | 144 | 1,5 | 5,436 | −19 | 361 | 6,0 | 5,470 | −39 | 1520 | 5,5 |
| 10 | 5,47 | +12 | 144 | 1,5 | 5,428 | −11 | 121 | 6,0 | 5,395 | +36 | 1300 | 5,5 |
| \(\Sigma\ 1100\) | \(\Sigma\ 6959\) | \(\Sigma\ 8853\) |
\[ S_{\mathrm{avg.}} = 5,59,\quad t_{\mathrm{avg.}} = 1,8 \]
\[ m_s = \pm 0,11\ (2,0\%) \]
\[ S_{\mathrm{avg.}} = 5,417,\quad t_{\mathrm{avg.}} = 6,4 \]
\[ m_s = \pm 0,028\ (0,56\%) \]
\[ S_{\mathrm{avg.}} = 5,431,\quad t_{\mathrm{avg.}} = 5,8 \]
\[ m_s = \pm 0,031\ (0,58\%) \]
Table 5
Determination of the area of figure 3
| Measurement Nos. | \multicolumn{4}{c}{By planimeter} | \multicolumn{4}{c}{By parallel palette} | \multicolumn{4}{c}{By ruler-palette} |
|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| Measurement Nos. | area \(S,\ \mathrm{cm}^2\) | \(\varepsilon_s\) | \(\varepsilon_s\varepsilon_s\) | \(t,\ \mathrm{min}\) | area \(S,\ \mathrm{cm}^2\) | \(\varepsilon_s\) | \(\varepsilon_s\varepsilon_s\) | \(t,\ \mathrm{min}\) | area \(S,\ \mathrm{cm}^2\) | \(\varepsilon_s\) | \(\varepsilon_s\varepsilon_s\) | \(t,\ \mathrm{min}\) |
| 1 | 7,47 | +4 | 16 | 2,0 | 7,792 | −16 | 256 | 6,5 | 7,795 | −9 | 81 | 6,0 |
| 2 | 7,47 | +4 | 16 | 2,0 | 7,752 | +24 | 576 | 6,5 | 7,850 | −64 | 4100 | 5,5 |
| 3 | 7,66 | −15 | 225 | 1,5 | 7,800 | −24 | 576 | 6,0 | 7,775 | +11 | 121 | 5,5 |
| 4 | 7,56 | −4 | 16 | 2,0 | 7,740 | +36 | 1290 | 6,0 | 7,780 | +6 | 36 | 5,5 |
| 5 | 7,56 | −4 | 16 | 1,5 | 7,708 | +68 | 4600 | 6,0 | 7,825 | −39 | 1520 | 5,5 |
| 6 | 7,75 | −24 | 576 | 2,0 | 7,760 | +16 | 256 | 6,0 | 7,785 | +1 | 1 | 6,5 |
| 7 | 7,47 | +4 | 16 | 1,5 | 7,816 | −40 | 1600 | 6,0 | 7,795 | −9 | 81 | 6,5 |
| 8 | 7,36 | +15 | 225 | 1,5 | 7,820 | −44 | 1940 | 6,5 | 7,740 | +46 | 2116 | 6,0 |
| 9 | 7,36 | +15 | 225 | 1,5 | 7,812 | −36 | 1290 | 6,5 | 7,750 | +36 | 1300 | 6,0 |
| 10 | 7,47 | −4 | 16 | 1,5 | 7,760 | +16 | 256 | 6,0 | 7,770 | +16 | 254 | 6,0 |
| | | | \(\Sigma\ 1347\) | | | | \(\Sigma\ 12640\) | | | | \(\Sigma\ 9610\) | |
| | \multicolumn{4}{c}{\(S_{\mathrm{avg}}=7,51\quad t_{\mathrm{avg}}=1,7\)} | \multicolumn{4}{c}{\(S_{\mathrm{avg}}=7,796\quad t_{\mathrm{avg}}=6,2\)} | \multicolumn{4}{c}{\(S_{\mathrm{avg}}=7,786\quad t_{\mathrm{avg}}=5,9\)} |
| | \multicolumn{4}{c}{\(m_S=\pm 0,12\ (1,6\%)\)} | \multicolumn{4}{c}{\(m_S=\pm 0,037\ (0,48\%)\)} | \multicolumn{4}{c}{\(m_S=\pm 0,033\ (0,42\%)\)} |
Table 6
Determination of the area of figure 4
| Measurement Nos. | \multicolumn{4}{c}{By planimeter} | \multicolumn{4}{c}{By parallel palette} | \multicolumn{4}{c}{By ruler-palette} |
|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| Measurement Nos. | area \(S,\ \mathrm{cm}^2\) | \(\varepsilon_s\) | \(\varepsilon_s\varepsilon_s\) | \(t,\ \mathrm{min}\) | area \(S,\ \mathrm{cm}^2\) | \(\varepsilon_s\) | \(\varepsilon_s\varepsilon_s\) | \(t,\ \mathrm{min}\) | area \(S,\ \mathrm{cm}^2\) | \(\varepsilon_s\) | \(\varepsilon_s\varepsilon_s\) | \(t,\ \mathrm{min}\) |
| 1 | 93,2 | +10 | 100 | 3,0 | 93,560 | −1,4 | 2 | 12,0 | 93,32 | −3 | 9 | 7,0 |
| 2 | 93,2 | +10 | 100 | 3,0 | 93,536 | +1,0 | 1 | 11,0 | 93,15 | +14 | 196 | 7,0 |
| 3 | 93,4 | −10 | 100 | 3,0 | 93,750 | −20,4 | 415 | 11,0 | 93,43 | −14 | 196 | 7,0 |
| 4 | 93,1 | +20 | 400 | 3,0 | 93,614 | −6,8 | 46 | 11,5 | 93,18 | +11 | 121 | 6,5 |
| 5 | 93,3 | 0 | 0 | 2,5 | 93,568 | −2,2 | 5 | 11,5 | 93,51 | −22 | 484 | 6,5 |
| 6 | 93,3 | 0 | 0 | 2,5 | 93,474 | +7,2 | 52 | 10,5 | 93,16 | +13 | 169 | 6,0 |
| 7 | 93,4 | −10 | 100 | 2,5 | 93,430 | +11,6 | 135 | 10,5 | 93,16 | +13 | 169 | 5,0 |
| 8 | 93,1 | +20 | 400 | 3,0 | 93,422 | +12,4 | 154 | 10,5 | 93,25 | +4 | 16 | 6,0 |
| 9 | 93,5 | −20 | 400 | 2,5 | 93,658 | −11,2 | 126 | 10,5 | 93,36 | −7 | 49 | 5,0 |
| 10 | 93,5 | −20 | 400 | 3,0 | 93,446 | +10,0 | 100 | 10,0 | 93,39 | −10 | 100 | 6,0 |
| | | | \(\Sigma\ 2000\) | | | | \(\Sigma\ 1036\) | | | | \(\Sigma\ 1509\) | |
| | \multicolumn{4}{c}{\(S_{\mathrm{avg}}=93,3\quad t_{\mathrm{avg}}=2,8\)} | \multicolumn{4}{c}{\(S_{\mathrm{avg}}=93,546\quad t_{\mathrm{avg}}=10,9\)} | \multicolumn{4}{c}{\(S_{\mathrm{avg}}=93,291\quad t_{\mathrm{avg}}=6,2\)} |
| | \multicolumn{4}{c}{\(m_S=\pm 0,14\ (0,16\%)\)} | \multicolumn{4}{c}{\(m_S=\pm 0,12\ (0,13\%)\)} | \multicolumn{4}{c}{\(m_S=0,13\ (0,14\%)\)} |
Table 7
Determination of the area of figure 5
| No. of meas. | Planimeter: area \(S,\ \mathrm{cm}^2\) | Planimeter: \(\varepsilon_S\) | Planimeter: \(\varepsilon_S\varepsilon_S\) | Planimeter: \(t,\ \mathrm{min}\) | Parallel palette: area \(S,\ \mathrm{cm}^2\) | Parallel palette: \(\varepsilon_S\) | Parallel palette: \(\varepsilon_S\varepsilon_S\) | Parallel palette: \(t,\ \mathrm{min}\) | Ruler-palette: area \(S,\ \mathrm{cm}^2\) | Ruler-palette: \(\varepsilon_S\) | Ruler-palette: \(\varepsilon_S\varepsilon_S\) | Ruler-palette: \(t,\ \mathrm{min}\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 89,4 | −10 | 100 | 3,0 | 89,772 | +0,9 | 1 | 11,5 | 89,60 | −2 | 4 | 6,5 |
| 2 | 89,00 | +30 | 900 | 2,5 | 89,738 | +4,3 | 18 | 11,5 | 89,74 | −16 | 256 | 6,5 |
| 3 | 89,4 | −10 | 100 | 3,5 | 89,632 | +14,9 | 222 | 10,5 | 89,67 | −9 | 81 | 6,5 |
| 4 | 89,5 | −20 | 400 | 2,5 | 89,782 | −0,1 | 0 | 11,0 | 89,84 | −26 | 676 | 6,5 |
| 5 | 89,6 | −30 | 900 | 2,5 | 89,916 | −13,5 | 182 | 10,5 | 89,3 | +28 | 784 | 6,0 |
| 6 | 89,4 | −10 | 100 | 4,0 | 89,767 | +1,4 | 2 | 10,5 | 89,49 | +9 | 81 | 6,0 |
| 7 | 89,0 | +30 | 900 | 3,0 | 89,896 | −11,5 | 132 | 10,5 | 89,66 | −8 | 64 | 6,0 |
| 8 | 89,2 | +10 | 100 | 2,5 | 89,658 | +12,3 | 150 | 10,0 | 89,63 | −5 | 25 | 6,0 |
| 9 | 89,1 | +20 | 400 | 2,5 | 89,800 | −1,9 | 4 | 10,0 | 89,39 | +19 | 361 | 6,5 |
| 10 | 89,3 | 0 | 0 | 2,5 | 88,850 | −6,9 | 48 | 10,0 | 89,47 | +11 | 121 | 6,5 |
| \(\Sigma\ 3900\) | \(\Sigma\ 759\) | \(\Sigma\ 2453\) | ||||||||||
| \(S_{\mathrm{avg.}}=89,3\) | \(t_{\mathrm{avg.}}=2,8\) | \(S_{\mathrm{avg.}}=89,781\) | \(t_{\mathrm{avg.}}=10,6\) | \(S_{\mathrm{avg.}}=89,58\) | \(t_{\mathrm{avg.}}=6,3\) | |||||||
| \(m_S=\pm0,21\ (0,23\%)\) | \(m_S=\pm0,09\ (0,1\%)\) | \(m_S=\pm0,16\ (0,18\%)\) |
Table 8
Determination of the area of figure 6
| No. of meas. | Planimeter: area \(S,\ \mathrm{cm}^2\) | Planimeter: \(\varepsilon_S\) | Planimeter: \(\varepsilon_S\varepsilon_S\) | Planimeter: \(t,\ \mathrm{min}\) | Parallel palette: area \(S,\ \mathrm{cm}^2\) | Parallel palette: \(\varepsilon_S\) | Parallel palette: \(\varepsilon_S\varepsilon_S\) | Parallel palette: \(t,\ \mathrm{min}\) | Ruler-palette: area \(S,\ \mathrm{cm}^2\) | Ruler-palette: \(\varepsilon_S\) | Ruler-palette: \(\varepsilon_S\varepsilon_S\) | Ruler-palette: \(t,\ \mathrm{min}\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 73,6 | −11 | 121 | 2,5 | 74,084 | +1 | 1 | 7,5 | 74,11 | −13 | 169 | 6,5 |
| 2 | 73,2 | +29 | 840 | 2,5 | 74,012 | +8 | 64 | 7,5 | 74,22 | −24 | 576 | 6,5 |
| 3 | 73,4 | +9 | 81 | 2,5 | 74,250 | −16 | 256 | 7,0 | 74,23 | −25 | 625 | 6,0 |
| 4 | 73,5 | −1 | 1 | 2,0 | 73,910 | +18 | 324 | 7,5 | 74,10 | −12 | 144 | 6,0 |
| 5 | 73,6 | −11 | 121 | 2,5 | 73,964 | +13 | 169 | 7,5 | 73,93 | +5 | 25 | 6,0 |
| 6 | 73,4 | +9 | 81 | 2,5 | 74,176 | −9 | 81 | 7,0 | 74,16 | −18 | 324 | 6,0 |
| 7 | 73,7 | −21 | 441 | 2,5 | 74,072 | +2 | 4 | 7,0 | 73,59 | +39 | 1520 | 6,0 |
| 8 | 73,6 | −11 | 121 | 2,5 | 74,196 | −10 | 100 | 7,0 | 74,07 | −9 | 81 | 6,0 |
| 9 | 73,6 | −11 | 121 | 2,0 | 74,072 | +2 | 4 | 7,0 | 73,63 | +35 | 1225 | 5,5 |
| 10 | 73,3 | +19 | 361 | 2,0 | 74,196 | −10 | 100 | 7,0 | 73,81 | +17 | 289 | 5,5 |
| \(\Sigma\ 2289\) | \(\Sigma\ 1103\) | \(\Sigma\ 4978\) | ||||||||||
| \(S_{\mathrm{avg.}}=73,49\) | \(t_{\mathrm{avg.}}=2,3\) | \(S_{\mathrm{avg.}}=74,093\) | \(t_{\mathrm{avg.}}=7,2\) | \(S_{\mathrm{avg.}}=73,985\) | \(t_{\mathrm{avg.}}=6,0\) | |||||||
| \(m_S=\pm0,16\ (0,22\%)\) | \(m_S=\pm0,11\ (0,15\%)\) | \(m_S=\pm0,23\ (0,32\%)\) |
Table 9
| No. | Area interval \(S,\ \mathrm{cm}^2\) | Error in determining \(S\), \(m_{S\%}\): by planimeter | Error in determining \(S\), \(m_{S\%}\): by parallel palette | Error in determining \(S\), \(m_{S\%}\): by ruler-palette, obtained theoretically | Error in determining \(S\), \(m_{S\%}\): by ruler-palette, obtained experimentally | Limiting error \(m_{S\%}\) of the proposed instrument | Average duration of one determination: by planimeter | Average duration of one determination: by parallel palette | Average duration of one determination: by ruler-palette |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1.8–10.0 | 1.8 | 0.5 | 0.7 | 0.5 | 2.1 | 1.8 | 5.8 | 5.6* |
| 2 | 10.0–100 | 0.2 | 0.12 | 0.45 | 0.2 | 1.5 | 2.6 | 9.6 | 6.2 |
Note: The duration of area determination by planimeter is given without taking into account the time needed to set up the instrument.
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In a one-time (not mass) determination of area on a plan, the proposed method (instrument) has an advantage over the planimeter.
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The procedure for measuring area by the described method is very simple.
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The instrument can easily be manufactured industrially, and its cost is not high.
In conclusion, it should be noted that the proposed instrument, manufactured industrially, can be successfully used by specialists, students, and pupils engaged in determining areas on plans in those cases where use of a planimeter is impossible.
In addition, the ruler-palette may find broad application under field conditions—for example, in geological and engineering surveys, land-reclamation work, in determining areas in agriculture, on tourist hikes, expeditions, and so on. The comparatively small increase in the time spent on determining area with the new instrument should be fully compensated by the very simple design of the instrument, its ease of use, and the possibility of using it under any conditions.