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Chapter 13 of 16

Planimeter & Pantograph

In the MP Vyapam Sub Engineer Civil syllabus under Surveying · 2 parts

Computation of Areas & Volumes · Maps, Scales & Map Preparation

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📑 Contents (20 sections)

Part 1 of 2

Computation of Areas & Volumes

Last reviewed 16 Sept 2026 · 8 min read

Areas from field measurements

Regular figures

Triangle: $A = \frac{1}{2} bh$ , or Heron's formula $A = s (s - a) (s - b) (s - c)$ , $s = \frac{a + b + c}{2}$ , or $A = \frac{1}{2} ab sin C$ .
Trapezium: $A = \frac{( a + b )}{2} h$ .
Areas of chain surveys are computed by dividing the main framework into triangles, and the strips between survey lines and irregular boundaries by offsets.

Offsets to irregular boundaries

Offsets $O_{1}, O_{2}, \dots, O_{n}$ taken at a regular interval $d$ along a survey line:

FormulaRules for areas from offsets

Mid-ordinate rule: $A = d (h_{1} + h_{2} + \dots + h_{n - 1})$ , where $h$ = ordinates at mid-points of divisions

Average ordinate rule: $A = \frac{O _{1} + O _{2} + \dots + O _{n}}{n} \times L$ ( $L$ = length of base)

Trapezoidal rule:

A = d [\frac{O _{1} + O _{n}}{2} + O_{2} + O_{3} + \dots + O_{n - 1}]

Simpson's one-third rule (boundary assumed parabolic; odd number of ordinates, even number of divisions):

A = \frac{d}{3} [O_{1} + O_{n} + 4 (O_{2} + O_{4} + \dots) + 2 (O_{3} + O_{5} + \dots)]

Simpson's rule is more accurate for curved boundaries; if the number of ordinates is even, the last division is computed separately by the trapezoidal rule.
The trapezoidal rule assumes straight-line boundaries between offsets.

Areas from coordinates

FormulaCoordinate method (shoelace formula)

For a closed figure with vertices $(x_{1}, y_{1}), (x_{2}, y_{2}), \dots, (x_{n}, y_{n})$ in order:

A = \frac{1}{2} i = 1 \sum n (x_{i} y_{i + 1} - x_{i + 1} y_{i})

(with $x_{n + 1} = x_{1}$ , $y_{n + 1} = y_{1}$ ), or equivalently $A = \frac{1}{2} ∣ \sum x_{i} (y_{i + 1} - y_{i - 1}) ∣$ .

Areas from latitudes and departures

Double meridian distance (DMD) method:

Meridian distance of a line — perpendicular distance of its mid-point from the reference meridian.
DMD of the first line = its departure.
DMD of any line = DMD of the preceding line + departure of the preceding line + departure of the line itself.
DMD of the last line = its departure with opposite sign (check).

2 A = \sum (DMD \times latitude) \Rightarrow A = \frac{1}{2} \sum D M D \times L

The double parallel distance (DPD) method uses departures and latitudes interchanged.

Areas from plans

Counting squares — a transparent sheet of squares placed over the plan; full squares plus estimated fractions.
Division into triangles, trapezoids or strips measured on the plan.
Computational/digital — digitising the boundary in CAD/GIS software.
Planimeter — mechanical or digital instrument measuring area by tracing the boundary.

Amsler polar planimeter

FormulaPlanimeter

A = M (F - I \pm 10 N + C)

$F$ = final reading, $I$ = initial reading (each reading has 4 digits from the wheel, drum and vernier)
$N$ = number of complete revolutions of the dial (+ if the zero mark passes the index clockwise, − if anticlockwise)
$M$ = multiplying constant (area per revolution of the wheel) — set by the tracing arm length
$C$ = constant added only when the anchor point is inside the figure; $10 M C$ = area of the zero circle (circle traced without rotation of the wheel)

The tracing point is moved clockwise around the figure. A digital planimeter displays area directly.

Volumes

Cross-sections for earthwork

For roads, railways and canals, cross-sections are taken at intervals $d$ along the centre line; formation width $b$ , side slopes $n$ horizontal : 1 vertical.

FormulaAreas of cross-sections

Level section (ground level across): $A = (b + nh) h$

Two-level section (ground with transverse slope 1 in $m$ , depth $h$ at centre):

Side widths: $w_{1} = (\frac{b}{2} + nh) \frac{m}{m - n}$ , $w_{2} = (\frac{b}{2} + nh) \frac{m}{m + n}$
Area: $A = \frac{1}{n} \frac{( \frac{b}{2} + nh ) ^{2} m ^{2}}{m ^{2} - n ^{2}} - \frac{b ^{2}}{4 n}$

Three-level section (depths $h_{1}$ , $h$ , $h_{2}$ at left edge, centre and right edge of formation; side widths $w_{1}$ , $w_{2}$ ):

A = \frac{h}{2} (w_{1} + w_{2}) + \frac{b}{4} (h_{1} + h_{2})

Multi-level (irregular) sections: by the coordinate method.

Volume from cross-sections

FormulaEarthwork volume formulas

Trapezoidal (average end area) formula:

V = d [\frac{A _{1} + A _{n}}{2} + A_{2} + A_{3} + \dots + A_{n - 1}]

Prismoidal formula (odd number of sections):

V = \frac{d}{3} [A_{1} + A_{n} + 4 (A_{2} + A_{4} + \dots) + 2 (A_{3} + A_{5} + \dots)]

Prismoidal correction (difference between trapezoidal and prismoidal volumes) for level sections between two cross-sections with depths $h_{1}$ and $h_{2}$ :

C_{p} = \frac{d n}{6} (h_{1} - h_{2})^{2}

(subtracted from the trapezoidal volume.)

The trapezoidal formula generally overestimates volumes; the prismoidal formula is more accurate (exact for prismoids).
Curvature correction (Pappus' theorem) applies when the centre line is curved and sections are unsymmetrical.

Volume from spot levels (borrow pits)

The area is divided into squares (or rectangles) or triangles of equal area; depths of cut/fill are found at corners.

FormulaSpot level (unit area) method

Rectangles (each of area $A$ ):

V = \frac{A}{4} (\sum h_{1} + 2 \sum h_{2} + 3 \sum h_{3} + 4 \sum h_{4})

$h_{1}$ = depths at corners common to one rectangle; $h_{2}$ — common to two; $h_{3}$ — three; $h_{4}$ — four.

Triangles (each of area $A$ ):

V = \frac{A}{3} \sum n h

( $n$ = number of triangles sharing that corner.)

Volume from contours

Volumes of reservoirs, hills and large excavations are computed from areas enclosed by contours using the trapezoidal or prismoidal formula with the contour interval as $d$ (see Contouring).

Part 2 of 2

Maps, Scales & Map Preparation

Last reviewed 16 Sept 2026 · 9 min read

Plans and maps

A plan is a large-scale drawing of a small area (e.g. a building site) where the earth's curvature is irrelevant.
A map is a small-scale representation of a larger area of the earth on a plane, showing natural and artificial features, usually with a projection and grid.

Classification of maps

By scale

Category	Typical scales	Examples
Large-scale	1 : 500 to about 1 : 10 000	Building plans, estate plans, town plans, cadastral maps
Medium-scale	About 1 : 25 000 to 1 : 50 000	Topographic maps (toposheets), district planning
Small-scale	1 : 250 000 and smaller	State maps, atlases, wall maps

By purpose/content

Topographic maps (relief, drainage, settlements, communications, vegetation), cadastral maps (property boundaries, land records), engineering maps (project sites, routes), contour maps, thematic maps (geology, soils, land use, population, rainfall), road and railway maps, navigation charts, city guide maps.

Choice of scale

The scale is chosen considering:

Purpose of the map and the detail required.
Size of the area and the paper/screen size.
Plotting accuracy — the smallest distance that can be plotted and read reliably on paper is about 0.25 mm; ground features smaller than $0.25 mm \times scale denominator$ cannot be shown to scale.
Time and cost of surveying (larger scales need more detail).

FormulaPlotting accuracy and scale

Minimum scale denominator \leq \frac{smallest ground feature to be plotted}{0.25 mm}

Survey of India topographic maps

The Survey of India (SOI) publishes topographic maps of the country.
Traditional sheet system: India was covered by 1 : 1 000 000 sheets of 4° × 4°; each divided into 16 sheets of 1° × 1° at 1 : 250 000 (e.g. sheet 53E); each degree sheet divided into 16 sheets of 15′ × 15′ at 1 : 50 000 (e.g. 53E/1 to 53E/16); each 1 : 50 000 sheet into 4 sheets of 7′30″ × 7′30″ at 1 : 25 000 (e.g. 53E/1/NW).
Under the National Map Policy (2005), SOI publishes two series:
- Defence Series Maps (DSMs) — for defence and national security (restricted).
- Open Series Maps (OSMs) — for public and development use, on the WGS 84 datum and UTM projection, with sensitive details excluded.
Digital topographic databases and online geoportals are increasingly used; India's geospatial guidelines and National Geospatial Policy (2022) liberalised access to geospatial data.

Map projections

A map projection is a systematic method of representing the curved surface of the earth (graticule of latitudes and longitudes) on a flat surface. No projection preserves all properties simultaneously.

Properties that may be preserved:

Conformal (orthomorphic) — shapes of small areas and angles preserved.
Equal-area (equivalent) — areas preserved.
Equidistant — distances preserved along certain lines.
Azimuthal — directions from a central point preserved.

Projection	Developable surface	Features / use
Cylindrical equal-area	Cylinder tangent at equator	Areas true; shapes distorted towards poles
Mercator	Cylinder (conformal)	Conformal; rhumb lines (constant bearing) are straight lines — navigation charts; large area distortion at high latitudes
Transverse Mercator	Cylinder tangent along a central meridian	Conformal; accurate in a narrow north–south zone — topographic mapping
Universal Transverse Mercator (UTM)	Secant transverse Mercator in 6° longitude zones	Scale factor 0.9996 at the central meridian; coordinates in metres with false easting of 500 000 m; widely used for GNSS and topographic maps (India spans several UTM zones)
Conical projections (e.g. Lambert conformal conic)	Cone tangent or secant along standard parallels	Suitable for mid-latitude regions extending east–west
Polyconic projection	Series of cones along each parallel	Used for Survey of India's older topographic sheets; minimal distortion for small sheets
Azimuthal (zenithal)	Plane tangent at a point	Polar maps; directions from centre true

Grids, graticules and references

Graticule — network of parallels (latitudes) and meridians (longitudes).
Grid — rectangular network of lines (eastings and northings) on a projected map, with numbered grid lines.
Grid reference — location given by eastings first, then northings ("read right, then up"): e.g. a six-figure reference such as 435782 means easting 43.5 and northing 78.2 grid units within the sheet's labelling.
Three norths: true north (geographic), grid north (direction of grid lines), magnetic north; the angles between them (grid convergence and magnetic declination) are shown in the map margin.

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