Area & partitioning
Converting a computed area between square feet, acres and hectares
One parcel area expressed in every unit a plat may call for, and the 4 ppm consequence of the two American feet made explicit — because 2 ppm in a length is 4 ppm in an area.
Given
- A parcel area was computed by coordinates as 297,109.70 ft². The coordinates came from a State Plane zone published in US survey feet, so the figure is 297,109.70 square US survey feet.
- Exact definitions available: 1 US survey foot = 1200/3937 m; 1 international foot = 0.3048 m; 1 acre = 43,560 ft² = 10 square chains; 1 chain = 66 ft; 1 hectare = 10,000 m².
- The plat must show the area in acres, and a companion metric exhibit must show it in hectares.
Required
- The area in acres and in square chains.
- The area in square meters and hectares, correctly reduced.
- The numerical consequence of applying the wrong foot, stated in the units of the answer.
Work it through yourself before reading on — the solution below shows every step, so there is no way to skim it without giving the answer away.
Worked solution
The two feet, and why an area feels the difference twice
Two feet have been in legal use in the United States. The international foot is 0.3048 m exactly; the US survey foot is 1200/3937 m exactly, which is 0.30480060960122 m. Their ratio is exactly 1.000002 to seven figures: the survey foot is longer by 2 parts per million.
An area is a length squared, so the ratio between the two squared feet is 1.000002², which is 1.000004 — 4 parts per million. That doubling is the whole point of this problem. A surveyor who has satisfied himself that 2 ppm is negligible in a distance has not thereby satisfied himself about the area.
Where the number matters is at scale. Four parts per million of a suburban lot is a fraction of a square foot. Four parts per million of a thousand-acre ranch is 0.004 acre, or 174 ft² — small, but a real and avoidable difference between two plats of the same ground.
1 US survey foot / 1 international foot = (1200/3937) / 0.3048 = 1.000002000004
difference in length: 2.000004 ppm
difference in area: 1.000002² = 1.000004000012 → 4.000012 ppm
on 1000 acres: 1000 × 43,560 × 4.0 × 10⁻⁶ = 174.24 ft² = 0.004 acAcres and square chains — no foot needed
The acre is defined as 43,560 ft², and the chain as 66 ft. Both sides of each ratio are squared in whatever foot the area is already in, so the foot cancels and the conversion is exact regardless of which foot was used. This is the one part of the problem where naming the foot does not matter.
Run both conversions and let them check each other: an acre is exactly 10 square chains, so the square-chain figure must be exactly ten times the acreage. This is free, and it catches a mistyped divisor immediately.
acres = 297,109.70 / 43,560 = 6.820700 ac
square chains = 297,109.70 / 66² = 297,109.70 / 4356 = 68.2070 sq ch
68.2070 / 10 = 6.82070 ac ✓
reported: 6.821 acresSquare meters and hectares — where the foot decides
Going metric means multiplying by the square of a foot in meters, and now the choice of foot is unavoidable. The coordinates came from a zone published in US survey feet, so the square US survey foot, 0.0929034116 m², is the correct multiplier.
Divide the square meters by 10,000 to get hectares. Report to 0.001 hectare, which is 10.8 ft² — a little finer than 0.001 acre, so a hectare figure carrying three decimals is honest for a parcel this size.
Do the wrong-foot reduction too, so the size of the error is on record rather than assumed.
1 US survey foot² = (1200/3937)² = 0.0929034116 m²
1 international foot² = 0.3048² = 0.0929030400 m²
correct: 297,109.70 × 0.0929034116 = 27,602.505 m² = 2.760251 ha
wrong foot: 297,109.70 × 0.0929030400 = 27,602.395 m² = 2.760239 ha
difference = 0.110 m² = 1.19 ft², i.e. 4 ppmCollect the answer in every unit
Put all the expressions of the same area side by side, each rounded to the precision the unit is normally reported in. Every line here describes the same ground; only the units differ.
The row worth pausing on is the second: the same physical area, re-expressed in square international feet, is 297,110.89 — 1.19 ft² larger than the same area in square US survey feet, because the international foot is the shorter unit and it takes more of them to cover the ground.
Name the foot on the face of the plat whenever coordinates or metric equivalents appear. 'Area: 6.821 acres' is unambiguous. '27,602.51 square meters' is not, unless the reader is told which foot got it there.
| Unit | Value | Note |
|---|---|---|
| Square US survey feet | 297,109.70 | as computed from the coordinates |
| Square international feet | 297,110.89 | same ground, shorter foot, 4 ppm more of them |
| Acres | 6.821 | 43,560 exact, foot-independent |
| Square chains | 68.207 | exactly 10 × acres, a free check |
| Square meters | 27,602.505 | via the square US survey foot |
| Hectares | 2.760 | square meters / 10,000 |
Answer
- Area = 6.821 acres, equivalently 68.207 square chains. The 43,560 ft² definition is exact and does not depend on which foot the square feet are in.
- Area = 27,602.505 m² = 2.760 hectares, reduced with the square US survey foot (0.0929034116 m²) because the coordinates came from a zone published in US survey feet.
- Using the international foot instead would give 27,602.395 m², an error of 0.110 m² or 1.19 ft² — 4 ppm, which is twice the 2 ppm by which the two feet differ in length.
- Expressed in square international feet the same area is 297,110.89 ft², against 297,109.70 square US survey feet.
Check
Round-trip the metric answer back to square feet: 2.760251 ha × 10,000 = 27,602.51 m², and 27,602.51 / 0.0929034116 = 297,109.8 square US survey feet, recovering the starting figure to about 0.1 ft².
Convert hectares straight to acres by a route that never touches feet: one hectare is 10,000 / (43,560 × 0.0929034116) = 2.471044 acres, so 2.760251 ha × 2.471044 = 6.820701 acres. That reproduces the direct division 297,109.70 / 43,560 = 6.820700 acres.
Check the 4 ppm claim numerically instead of trusting the algebra: 297,110.89 / 297,109.70 = 1.0000040, which is the square of 1.0000020 as expected.
Check the chain relationship: 66² = 4356, and 4356 × 10 = 43,560 exactly, so the square-chain and acre figures must stand in a ratio of exactly 10. 68.2070 / 6.820700 = 10.0000.
More area & partitioning
All in this category- Area of a closed parcel from coordinates by the shoelace ruleA five-corner parcel is given as plane coordinates. Work the cross-product sum term by term to square feet and acres, and use the sign of the result to prove the corner list is in boundary order.
- The same parcel by double meridian distanceThe HAWTHORN TRACT area recomputed on a DMD sheet — the form a plat's computation panel still shows — and set against the coordinate answer to prove the two methods are the same computation.
- Area of a parcel with one circular-arc boundaryA five-sided parcel whose southerly boundary is a circular arc. Compute the area of the chord polygon by coordinates, add the circular segment R²/2·(Δ − sin Δ), and check the segment as sector minus triangle.