How is the coordinate system set up for the offsets?

Chain line as x-axis, offsets as y-values, first offset position as origin

Offsets as x-axis and chain line as y-axis

North-south as x-axis using a compass

What closed figure is used to compute the area?

Chain-line ends plus the offset ends taken in order round the area

Only the offset ends joined together

A triangle from the first, middle and last offsets

A rectangle of length times maximum offset

Which formula is used for the area?

1/2 |sum (x_i y_(i+1) - x_(i+1) y_i)|

How does the answer verify the result?

By adding four trapezoidal strips, giving the same 38.479 m^2

By repeating with Simpson's rule only

By comparing with the perimeter

2069 Chaitra · 8 marks

Area by coordinate method from offsets

Short answer

Take the chain line as x-axis and form a closed polygon with the chain ends and offset endpoints. Coordinate area = ½|Σ(x_i y_{i+1} − x_{i+1} y_i)| = 38.479 m².

Calculate the area by the coordinate method from the following perpendicular offsets taken from a chain line to a boundary.

Chainage (m)	0.00	3.75	6.50	11.30	16.45
Offset (m)	1.45	2.50	2.95	2.10	2.35

Method. The chain line is taken as the x-axis and the first offset position as the origin. The ends of the offsets and the two chain-line corners form a closed polygon whose area is found from its coordinates:

\text{Area} = \frac{1}{2}\left|\sum \left(x_i\,y_{i+1} - x_{i+1}\,y_i\right)\right|

Coordinates (taken in order round the area)

Point	$x$ (m)	$y$ (m)
Chain start	0.00	0.00
Chain end	16.45	0.00
offset 5	16.45	2.35
offset 4	11.30	2.10
offset 3	6.50	2.95
offset 2	3.75	2.50
offset 1	0.00	1.45

Calculation

Pair	$x_i y_{i+1}$	$x_{i+1} y_i$	Difference
P1→P2	0.0000	0.0000	0.0000
P2→P3	38.6575	0.0000	38.6575
P3→P4	34.5450	26.5550	7.9900
P4→P5	33.3350	13.6500	19.6850
P5→P6	16.2500	11.0625	5.1875
P6→P7	5.4375	0.0000	5.4375
P7→P1	0.0000	0.0000	0.0000
Sum			76.9575

\text{Area} = \frac{1}{2}\times |76.9575| = 38.4788\ \text{m}^2

Check (trapezoidal strips): the four strips give 7.4063 + 7.4938 + 12.1200 + 11.4588 = 38.4788 m².

Answer: area = 38.479 m².

✓ Answer checked by Claude Opus 5.5 (high effort)

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Area by coordinate method from offsets

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