Area & Measurement Problems
~10 min read · Convert acres and square feet and price land by front foot and per acre.
Area problems are unit fluency: 43,560 square feet to the acre, 640 acres to the section, and the geometry of rectangles and triangles priced per square foot or front foot. The numbers never change — memorize three constants and the category surrenders.
The constants
1 acre = 43,560 sq ft. 1 section = 1 square mile = 640 acres; a township (6×6 miles) holds 36 sections. 1 mile = 5,280 ft; a square mile is 5,280 × 5,280 = 27,878,400 sq ft. Rectangular-survey fractions divide the section: a quarter-section = 160 acres; 'the NW¼ of the SE¼' = 640 × ¼ × ¼ = 40 acres — multiply the denominators through.
- 43,560 sq ft/acre — the king constant
- 640 acres/section; 36 sections/township
- Stacked fractions multiply: ¼ of ¼ of a section = 40 acres
Geometry in use
Rectangle: length × width. Triangle: ½ × base × height. Irregular lots decompose into rectangles + triangles. Volume (concrete, fill): L × W × depth — with the depth converted to feet FIRST (6 inches = 0.5 ft), and cubic yards = cubic feet ÷ 27. Frontage pricing: '$1,500 per front foot' pays on the street-facing dimension only — the first number in '80 × 120' by convention.
- Triangles halve; decompose the odd shapes
- Inches → feet before multiplying; ÷27 for cubic yards
- Front-foot price × frontage, ignore depth
Cost per unit conversions
Price per sq ft = price ÷ area; per acre = price ÷ (area ÷ 43,560). Comparing a $217,800 five-acre parcel to a $1.10/sq-ft parcel means converting one side: 5 ac = 217,800 sq ft, so $1.00/sq ft — cheaper. The exam mixes units deliberately; convert to ONE unit before comparing.
Worked example
A buyer considers 'the S½ of the NE¼ of Section 12' at $9,000 per acre, and separately a rectangular corner lot 90 ft × 145 ft listed at $2,610 per front foot on its 90-ft street side. Compute both prices — and the corner lot's cost per square foot.
The survey parcel: 640 × ¼ = 160; × ½ = 80 acres. Price = 80 × 9,000 = $720,000. The corner lot: front-foot pricing pays on the 90-ft frontage → 90 × 2,610 = $234,900. Its area = 90 × 145 = 13,050 sq ft, so per-square-foot cost = 234,900 ÷ 13,050 = $18.00/sq ft. Every move was a memorized constant or a two-step multiplication: denominators through for the survey fraction, frontage-only for the front-foot price, area division for the unit cost. Practice until 43,560 and 640 surface without thought.
Common exam pitfalls
Misreading stacked survey fractions.
Work outside-in multiplying denominators: the S½ of the NE¼ = 640 ÷ 4 ÷ 2 = 80 acres.
Pricing front feet by area.
Front-foot prices pay frontage × rate — depth is irrelevant to that number.
Multiplying inches into square-foot math.
Convert every dimension to feet first; 8 inches is 0.667 ft, not 8.
43,560 in the acre, 640 in the section, half-base-height for triangles — convert first, multiply second.
Recap
- 1 acre = 43,560 sq ft; 1 section = 640 acres; township = 36 sections
- Survey descriptions: multiply the fraction denominators through
- Rectangle L×W; triangle ½bh; decompose irregulars
- Volume in consistent feet; ÷27 to cubic yards
- Front-foot pricing uses frontage only
- Convert to one unit before any comparison
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