Offset calculation exercise

Calculate the area and perimeter of the Lot

A chain line (line & figures = red) is set up across an existing polygonal shaped lot running from Station A to Station B with given measurements along the chain line as well as the offsets (lines & figures = green) measurements to all other corner pegs of the lot.

From the given survey data find the area in square metre (three decimals) of the lot and the perimeter in mm (running metre of fencing).

As can be seen all single areas a to i consist of triangle and Trapeziums.

To calculate the area you only need to know the following formula:

(a) triangles and (b) trapeziums

All offset measurements are perpendicular to the chaine line

Figure 1

The area of    Triangle a is 17.202 × 22.183/2 = 190.796 m²

Triangle b is 15.199 × 18.999/2 = 144.383 m

The area of    Trapezium c is (30.091 + 22.183)/2 × (32.350 - 17.202) = 395.923 m²

Trapezium d is (29.064 + 18.999)/2 × (38.410 - 15.199 = 557.795 m

Finish the calculation for the rest of the areas (answer = 3,212 m²)

The calculation of the perimeter of the lot is more cumbersome. You may use either Pythagoras theorem or the tragicomic functions.


length of  a1 = (17.202 + 22.183) = 28.071 m

b1 = (15.199 + 18.999) = 24.330 m

c1 = [(32.350 -17.202) + (30.091 - 22.183)] = 17.088 m

d1 = [(38.410 -115.199) + (29.064 - 15.199)] = 18.783 m

Finish the calculation for the rest of the boundary sites (answer = 210.707 m)

Trigonometric functions

To use trigonometric function to calculate the perimeter is too cumbersome as no angles given.
Practical Project 2 (rectangular coordinates)

Setting out a perpendicular line using builders triangle 3, 4, & 5 instead with an optical square.

Figure 2
  This method is applicable for Project 2 to set out wooden pegs for a L-shape building.

Set up the chainline starting at Station A 0.000 and finished at Station C 16.628. Peg out these station and all the stations between. Record all peg dimension in your field notes as shown in Figure 3.

From Station 0.000 create a perpendicular offset and place a peg at A distance 3.001. The next off-set is at chainline peg 3.765 at a distance to F of 5.721. Repeat this process for all off-set positions at B, C, D and E.

Put a string line around all pegs to show the building outline.

Practical Project 3 (polar coordinates)

The method of polar coordinates is used for Project 3. This project requires the theodolite, which must be booked in advance (only one is available).

Figure 3
Place a pole peg at an appropriate position that will not interfere with any obstruction at your set-out location. Set up the theodolite above the centre of the pole peg using the optical plumb.

Position a wooden peg at A at a distance of 10497. Zero the instrument on this position. Measure an angle of 87.650° and place another peg at at a distance of 7232 from the centre of the pole peg as shown in Figure 3. Repeat this process for all other polar positions at B, C, D, E and F. Check whether the reading for A finished at 360°.

Put a string line around all pegs to show the building outline.

In order to describe the position of a point, two coordinates are required. Polar coordinates need a line and an angle. Rectangular (Cartesian) coordinates need two points within an orthogonal system. The total station measures polar coordinates; these are recalculated as Cartesian coordinates within the given orthogonal system, either within the instrument itself or subsequently in the office.

Figure 4

Click here for a comparison of polar and rectangular coordinates.

Working with Range Poles

A range pole (also called a lining rod) is a metal or wood pole, usually about 2 metre long and about 20 mm in diameter; it is provided with a steel point or shoe and painted in alternate bands of red and white to increase its visibility. The range pole is held vertically on a point or plumbed over a point, so the point may be observed through an optical instrument. It is primarily used as a sighting rod for either linear or angular measurements.

Setting out straight lines over a short or long distance
Figure 5

Optical square

A small hand instrument used in setting off a right angle. The optical square has two plane mirrors placed at an angle of 45 to each other. Two range poles are placed on a chain line at position A and B. Range pole A should be seen in the lower mirror, and the range pole B in the upper mirror. Move range pole C (middle slot) in left or right direction until its lined up with the range pole in the lower and upper mirror. The lines to the point of observation from the two observed objects will then meet in a right angle.
( In another form of optical square, a single plane mirror makes an angle of 45 with a sighting line. it used for sitting out a right angle.)

Figure 6

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