Area of Land & the Trapezoidal Rule (incl. Rainfall V=Ah)
Learn the trapezoidal rule for the area of an irregular block of land in NSW Year 12 Mathematics Standard 2. You measure equally-spaced perpendicular offsets from a survey baseline, treat each strip as a trapezium, and add the strip areas — adding the outer offsets once and doubling every offset in between.
You will also use \(V=Ah\) to find the volume of rainfall collected over a block, converting the rain depth from millimetres to metres and the result to litres or hectares. These are core Standard 2 skills for site plans, aerial photos, paddocks and reservoirs.
Theory
The trapezoidal rule estimates the area of an irregular block of land from equally-spaced perpendicular offsets, and \(V=Ah\) gives the volume of rain collected over an area. This Year 12 Standard 2 (NSW) guide shows the single-strip and several-strip formulas, how to double only the middle offsets, and how to convert between \(\text{mm}\), litres and hectares.
The trapezoidal rule estimates the area of an irregular block of land whose boundary is not a simple shape. A straight survey baseline is drawn, and perpendicular offsets (the widths to the boundary) are measured at equal spacings. Each strip between two offsets is treated as a trapezium.
The equal spacing between offsets is the strip width \(h\). For a single strip the area is \(A=\tfrac{h}{2}(d_f+d_l)\); across several strips you add the two outer offsets once and double every offset in between. It is an estimate because the true boundary is usually curved.
Once the area \(A\) is known, \(V=Ah\) gives the volume of rainfall collected over that area, where \(h\) is the rain depth in metres. This is a core Year 12 Mathematics Standard 2 (NSW) skill for site plans, aerial photos and reservoir problems.
For a single strip with end offsets \(d_f\) and \(d_l\) a strip width \(h\) apart:
For several strips, add the outer offsets once and double each middle offset:
For the volume of rainfall over an area \(A\), with rain depth \(h\) in metres:
How to find the area and rainfall volume
- List the offsets in order along the baseline and note the strip width \(h\) (the equal spacing).
- Add the outer offsets once — the first and the last.
- Double every middle offset and add these to the total inside the brackets.
- Multiply by \(\tfrac{h}{2}\) to get the area \(A\) in \(\text{m}^2\).
- For rainfall, convert the depth to metres and use \(V=Ah\); convert to litres or hectares if the question asks.
One strip, so \(A=\tfrac{h}{2}(d_f+d_l)\).
| \(A\) | \(=\) | \(\dfrac{h}{2}(d_f+d_l)\) |
| \(A\) | \(=\) | \(\dfrac{30}{2}(18+26)\) |
| \(A\) | \(=\) | \(15\times 44\) |
| \(A\) | \(=\) | \(660\text{ m}^2\) |
Two strips, so double the single middle offset.
| \(A\) | \(=\) | \(\dfrac{h}{2}\big[d_1+d_3+2d_2\big]\) |
| \(A\) | \(=\) | \(\dfrac{20}{2}\big[12+16+2(24)\big]\) |
| \(A\) | \(=\) | \(10\,(28+48)\) |
| \(A\) | \(=\) | \(760\text{ m}^2\) |
Convert the depth to metres, then use \(V=Ah\).
| \(h\) | \(=\) | \(24\text{ mm}=0.024\text{ m}\) |
| \(V\) | \(=\) | \(900\times 0.024\) |
| \(V\) | \(=\) | \(21.6\text{ m}^3\) |
| \(V\) | \(=\) | \(21.6\times 1000=21\,600\text{ L}\) |
The garden collects \(21.6\text{ m}^3\), or \(21\,600\text{ L}\).
Double the three middle offsets for the area, then apply \(V=Ah\).
| \(A\) | \(=\) | \(\dfrac{25}{2}\big[14+10+2(26+32+22)\big]\) |
| \(A\) | \(=\) | \(12.5\,(24+160)=2300\text{ m}^2\) |
| \(h\) | \(=\) | \(40\text{ mm}=0.04\text{ m}\) |
| \(V\) | \(=\) | \(2300\times 0.04=92\text{ m}^3\) |
The block is \(2300\text{ m}^2\) and the rain is \(92\text{ m}^3\).
Common pitfalls
Frequently asked questions
What is the trapezoidal rule for the area of land?
It estimates the area of an irregular block by splitting it into equal-width strips along a survey baseline and treating each strip as a trapezium. You add the two outer offsets once, double every offset in between, then multiply the total by half the strip width.
Do you double the first and last offset?
No. The first and last offsets are only counted once. Only the offsets in between them are doubled. A common mistake is to double every offset, which makes the area too large.
What is an offset in a land survey?
An offset is a perpendicular distance measured from the straight survey baseline out to the irregular boundary of the block. Offsets are taken at equal spacings, and that spacing is the strip width h.
How do you calculate the volume of rainfall on an area?
Use V equals A times h, where A is the area in square metres and h is the rain depth in metres. Convert the depth from millimetres to metres first by dividing by 1000, so 25 mm becomes 0.025 m.
How many litres are in a cubic metre?
One cubic metre is 1000 litres. So once you have the rainfall volume in cubic metres, multiply by 1000 to get litres. For example, 21.6 cubic metres is 21 600 litres.
How do you convert square metres to hectares?
Divide by 10 000, because one hectare is 10 000 square metres. So a 17 500 square metre paddock is 1.75 hectares.