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Land measurement from plans
Offset (traverse) surveys and field books, splitting a block into triangles and trapezia to find its area, converting to hectares, and using the trapezoidal rule with measurements taken from a scaled plan.
Land is rarely a neat rectangle. Surveyors deal with this by running one straight line through the block and measuring out from it at right angles. Those measurements cut the land into simple shapes whose areas you already know. The same idea, applied to a scaled plan, lets you estimate the area of a dam or a creek-side paddock without setting foot on it.
Revise: the trapezoidal rule for one strip is in Year 11 Area of real shapes and the trapezoidal rule.
Offset surveys
In an offset survey (also called a traverse survey) the surveyor walks a straight traverse line AB across the block. At each corner of the boundary, they record how far along AB they are and the offset: the perpendicular distance from AB out to the corner. The results go in a field book: distances along AB down the middle, read from the bottom up, with left offsets on the left and right offsets on the right.
| Left (m) | Along AB (m) | Right (m) |
|---|---|---|
| $80$ (B) | ||
| $30$ | $50$ | |
| $35$ | $20$ | |
| $25$ | $20$ | |
| $0$ (A) |
Each offset meets AB at a right angle. So between consecutive stations on one side you get either a right-angled triangle (at A or B, where the offset is $0$) or a trapezium with two parallel offsets.
Triangle: $A = \tfrac12 \times \text{base along AB} \times \text{offset}$
Trapezium: $A = \dfrac{\text{offset}_1 + \text{offset}_2}{2} \times \text{distance between them}$
$1$ ha $= 10{,}000$ m²
Find the area of the paddock in the field book above, in m² and in hectares.
On Paddock, press Next piece and check each area against Example 1. Before each press, say whether the next piece is a triangle or a trapezium and why. Then switch to Orchard, read the field book, and predict the total before stepping through. Which piece is the largest, and why?
A radial survey measures out from a central point instead, using distances and compass bearings. Its triangles are not right-angled, so it needs the area formula $\tfrac12 ab \sin C$, which is in the Non-right-angled trigonometry chapter.
The trapezoidal rule from a plan
For a curved boundary, such as a creek or a dam, draw equally spaced parallel lines across the plan, measure each width, and treat each strip as a trapezium. Using the rule once for each strip and adding gives a good estimate.
$A \approx \dfrac{h}{2}\left(d_f + d_l\right)$, where $h$ is the strip width and $d_f$, $d_l$ are the widths at each end.
A $1 : 1000$ plan of a dam has widths of $0$, $32$, $41$, $28$ and $0$ mm measured at intervals of $20$ mm. Estimate the dam's surface area using four applications of the trapezoidal rule.
On a $1 : 500$ plan, a creek-side block has widths of $46$, $52$ and $38$ mm, measured $40$ mm apart. Estimate its area with two applications of the trapezoidal rule.
Your turn
A field book along a $60$ m traverse AB has one left offset of $24$ m at $40$ m, and one right offset of $18$ m at $15$ m. Find the area of the block.
Show the working
Left: triangles $\tfrac12 \times 40 \times 24 = 480$ and $\tfrac12 \times 20 \times 24 = 240$, total $720$ m². Right: $\tfrac12 \times 15 \times 18 = 135$ and $\tfrac12 \times 45 \times 18 = 405$, total $540$ m². Area $= 720 + 540 = 1260$ m².
A right-angled triangle in an offset survey has a base of $20$ m along AB and an offset of $25$ m. What is its area, in m²?
$\tfrac12 \times 20 \times 25 = 250$ m².
Offsets of $25$ m and $30$ m are taken at $20$ m and $50$ m along AB, on the same side. What is the area of the trapezium between them, in m²?
Width $= 50 - 20 = 30$ m. $\dfrac{25 + 30}{2} \times 30 = 27.5 \times 30 = 825$ m².
In an offset survey, the piece between the last offset and B is:
At B the offset is $0$, so the piece has one perpendicular side and comes to a point: a right-angled triangle.
The pieces of a paddock are $250$, $825$, $450$, $350$ and $450$ m². What is the total area, in m²?
$250 + 825 + 450 + 350 + 450 = 2325$ m².
Write $2325$ m² in hectares.
$2325 \div 10{,}000 = 0.2325$ ha.
A radial survey needs a different area formula because:
Radial surveys measure distances and bearings from a central point, giving general triangles.
On a $1 : 1000$ plan, widths of $0$, $32$, $41$, $28$, $0$ mm are taken $20$ mm apart. Estimate the real area, in m², with four applications of the trapezoidal rule.
Real widths in m equal the mm values, and $h = 20$ m. $10(0 + 32) + 10(32 + 41) + 10(41 + 28) + 10(28 + 0) = 320 + 730 + 690 + 280 = 2020$ m².
On a $1 : 500$ plan, two widths are measured $40$ mm apart. What is the real strip width $h$, in metres?
$40 \times 500 = 20{,}000$ mm $= 20$ m.
Exam style: On a $1 : 500$ plan, widths of $46$, $52$ and $38$ mm are measured $40$ mm apart. Estimate the area, in m², using two applications of the trapezoidal rule.
Real widths: $23$, $26$, $19$ m, and $h = 20$ m. $10(23 + 26) + 10(26 + 19) = 490 + 450 = 940$ m².
Exam style: A $60$ m traverse AB has a left offset of $24$ m at $40$ m and a right offset of $18$ m at $15$ m. Find the area of the block, in m².
Left: $\tfrac12 \times 40 \times 24 + \tfrac12 \times 20 \times 24 = 480 + 240 = 720$. Right: $\tfrac12 \times 15 \times 18 + \tfrac12 \times 45 \times 18 = 135 + 405 = 540$. Total $= 1260$ m².