Land and building math guide
Area calculations with units that cancel cleanly
Start by writing the unit the question asks for. A lot can have 150 feet of frontage, 30,000 square feet of area and a price quoted per acre. Those numbers describe different things. Identify the shape, label its dimensions and match the units before calculating.
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The short answer
For a rectangle, multiply length by width. For a triangle, multiply base by perpendicular height and divide by two. For a circle, use pi times radius squared. Convert square feet to acres by dividing by 43,560; multiply acres by 43,560 to get square feet. One square yard is nine square feet. Use the formula table for other shapes, and keep the requested unit beside the answer.
Reviewed October 25, 2026. National I.B.4 and I.B.5 in the June 24, 2026 PSI broker outline cover measuring structures and land measurement. The whole Property Ownership domain has 10 of 100 scored national questions, with no separate area-calculations quota. This page teaches geometry and unit conversions for stated exam facts. It does not establish parcel boundaries or a building's reportable living, usable or rentable area.
Work from the shape to the requested unit
- Identify the requested result: frontage, perimeter, square feet, acres, a missing dimension or price. A length answer uses feet; a surface answer uses square feet or another area unit.
- Sketch the stated shape and label its measurements. Rectangle sides meet at right angles. A triangle or trapezoid needs perpendicular height. A circle needs radius, which is half its diameter.
- Make the units compatible. Convert yards to feet before multiplying by a dimension in feet, or calculate entirely in yards and convert the resulting square yards. Do not treat feet times yards as square feet.
- Choose the formula. Split a composite shape into nonoverlapping pieces or subtract an actual cutout from its enclosing shape. If the facts do not supply enough dimensions, do not invent a rectangle or a right angle.
- Convert the area if needed: square feet divided by 43,560 gives acres; square feet divided by nine gives square yards. Match a price per acre, square foot, square yard or front foot to that same kind of quantity.
- Retain precision until the requested rounding step, then check the size. A triangle with the same base and height as a rectangle has half its area. A circle fits inside a square whose side equals its diameter. A result labeled in feet alone is not an area. Rounding for one textbook exercise or program does not establish an exam-wide rounding rule.
| Problem | Formula or conversion | Exam safeguard |
|---|---|---|
| Rectangle | Length x width | Use compatible units |
| Square | Side x side | Area unit is squared |
| Triangle | Base x perpendicular height / 2 | Do not use sloped side |
| Circle | Pi x radius x radius | Diameter / 2 first |
| Trapezoid | (Parallel side 1 + side 2) / 2 x height | Height is perpendicular |
| Square feet to acres | Square feet / 43,560 | Moving to larger unit |
| Acres to square feet | Acres x 43,560 | Moving to smaller unit |
| Square yards to square feet | Square yards x 9 | Square the factor 3 |
| Square yards per acre | 4,840 square yards | 43,560 square feet / 9 |
| 1 chain | 66 feet or 4 rods | Linear survey unit |
| 1 rod | 16.5 feet | 320 rods per mile |
| Square chain | 4,356 square feet; ten per acre | 66 feet x 66 feet |
| Square mile | 640 acres | Exact area conversion; not a linear mile |
| Standard section | 640 nominal acres | Use supplied actual area for an irregular section |
| Survey township | 36 sections or 23,040 nominal acres | Actual survey can vary |
The distinctions that matter
| Distinction | How it works | What to notice |
|---|---|---|
| Linear foot vs. square foot | A linear foot measures one dimension. A square foot measures a surface one foot by one foot. | Boundary length versus surface area. |
| Frontage vs. area | Frontage is a boundary length. Area also requires depth or enough other geometry to measure a surface. | One dimension versus two-dimensional result. |
| Perimeter vs. area | Perimeter adds boundary lengths. Area measures the enclosed surface through a shape formula. | Distance around versus space inside. |
| Square yard vs. square foot | A square yard is 3 feet by 3 feet and therefore equals 9 square feet. | Square the linear conversion. |
| Mile vs. square mile | A mile is 5,280 linear feet. A square mile is one mile by one mile and contains 640 acres. | Length versus large area. |
| Radius vs. diameter | Radius runs from center to edge. Diameter crosses the circle through its center and equals two radii. | Half-width versus full-width. |
| Triangle side vs. perpendicular height | A boundary side can slope. Height is the perpendicular distance from the selected base to the opposite vertex. | Edge length versus right-angle distance. |
| Section vs. township | A nominal section contains 640 acres. A nominal survey township contains 36 sections or 23,040 acres. | One grid square versus 36-square grid. |
| Nominal acreage vs. surveyed acreage | Nominal acreage follows theoretical grid or shape math. Surveyed acreage reflects controlling evidence and actual professional measurement. | Exam grid value versus field result. |
| Gross area vs. net or usable area | Gross area follows its stated outer or total definition. Net or usable area excludes spaces under the applicable measurement definition. | Total envelope versus defined usable portion. |
Work through the facts
Convert a rectangular lot to acres
A rectangular parcel measures 330 feet by 264 feet. What area does it contain in square feet and acres?
- Rectangle area is 330 x 264 = 87,120 square feet.
- Convert to acres by dividing 87,120 by 43,560.
- The quotient is exactly 2.
Answer: The parcel contains 87,120 square feet, or 2 acres.
Find triangular acreage
A triangular tract has a 300-foot base and a 220-foot perpendicular height. Find area in acres to the nearest hundredth.
- Triangle area is 300 x 220 / 2 = 33,000 square feet.
- Convert by dividing 33,000 by 43,560.
- The result is approximately 0.7576 acre.
Answer: The triangular tract contains about 0.76 acre.
Calculate a circular parcel
A circular parcel has a diameter of 200 feet. Using calculator pi, what is its area to the nearest square foot?
- Radius is 200 / 2 = 100 feet.
- Area is pi x 100 squared, or about 31,415.93 square feet.
- Round only the final result to the nearest square foot.
Answer: The circle contains approximately 31,416 square feet.
Subtract a missing corner
An L-shaped lot fits inside a 200-foot by 150-foot rectangle, but a 50-foot by 40-foot corner is missing. What is the lot area?
- Outer rectangle is 200 x 150 = 30,000 square feet.
- Missing corner is 50 x 40 = 2,000 square feet.
- Net area is 30,000 - 2,000 = 28,000 square feet.
Answer: The L-shaped lot contains 28,000 square feet.
Recover a missing lot depth
A rectangular lot contains 21,780 square feet and has 110 feet of frontage. What is its depth?
- For a rectangle, depth equals area divided by frontage.
- 21,780 square feet / 110 feet = 198 feet.
- Square feet divided by feet leaves a linear-foot answer.
Answer: The lot depth is 198 feet.
Price acreage after a conversion
A parcel contains 130,680 square feet and sells at $42,000 per acre. What is the total land price?
- Acreage is 130,680 / 43,560 = 3 acres.
- Total price is 3 x $42,000.
- Matching acres with price per acre prevents a unit mismatch.
Answer: The total land price is $126,000.
Convert mixed dimensions before multiplying
A rectangular parcel measures 60 yards by 90 feet. Find its square footage.
- Convert 60 yards to 180 feet.
- Multiply 180 feet by 90 feet.
- Alternatively, convert 90 feet to 30 yards: 60 x 30 = 1,800 square yards, then multiply by nine.
Answer: The parcel contains 16,200 square feet.
Use a trapezoid's parallel boundaries
A trapezoidal lot has parallel sides of 150 and 250 feet. Their perpendicular separation is 180 feet. Find its area.
- Average the parallel sides: (150 + 250) / 2 = 200 feet.
- Multiply the average by perpendicular height: 200 x 180.
- A sloping side is not the stated height.
Answer: The lot contains 36,000 square feet.
Price a floor measured in square yards
A rectangular floor measures 18 feet by 15 feet. Flooring costs $36 per square yard. The question excludes waste, installation and other charges. What is the material price?
- Area is 18 x 15 = 270 square feet.
- Convert 270 / 9 = 30 square yards.
- Multiply 30 square yards by $36 per square yard.
Answer: The stated material price is $1,080.
An acre does not specify a shape
A rectangle measures 66 feet by 660 feet. A different parcel is square and contains one acre. Must their sides be the same?
- The rectangle has 66 x 660 = 43,560 square feet, exactly one acre.
- Only the square's side is the square root of 43,560, approximately 208.7103 feet.
- Equal areas do not require equal shapes, sides or perimeters.
Answer: No. Both contain one acre; the square's side is about 208.71 feet, while the rectangle's stated sides are 66 and 660 feet.
Mistakes to avoid
- Add length and width to find area. Adding sides helps find perimeter; rectangle area requires multiplication.
- Report a two-dimensional result in linear feet. Area must be labeled in square feet, square yards, acres, or another area unit.
- Convert square yards to square feet with a factor of 3. One square yard equals 9 square feet because both dimensions convert.
- Multiply square feet by 43,560 to get acres. Divide square feet by 43,560; multiply only when converting acres to square feet.
- Use the diameter as radius in circle area. Divide diameter by two before squaring the radius.
- Use a triangle's sloped edge as its height. Height must be perpendicular to the selected base.
- Add overlapping composite shapes without adjustment. Choose nonoverlapping pieces or subtract the overlap once.
- Mix feet and yards in the same multiplication. Convert both dimensions to one linear unit before calculating area.
- Call one linear mile 640 acres. One square mile contains 640 acres; one mile alone measures length.
- Assume every surveyed section contains exactly 640 acres. The section figure is nominal; use actual acreage when the problem supplies it.
- Round the first shape before adding an irregular parcel. Retain precision through every component and round the final requested result.
- Treat a sketch calculation as a controlling survey. Exam geometry is simplified; actual boundaries and area require controlling evidence and professional survey work.
- Every one-acre lot has sides of about 208.71 feet. That side length applies to a square acre. An acre measures area, not shape.
- Frontage and total area are interchangeable pricing bases. A price per front foot multiplies frontage. A price per square foot multiplies area. Read the actual quoted basis.
- The entire outside rectangle is automatically reportable living area. Geometry supplies an area for stated dimensions. The applicable measurement definition determines which spaces count in the requested building metric.
Check your understanding
These are original teaching questions, not recalled or predicted PSI questions. Answer first, then explain why the other choices fail.
1. How many square feet are in 1.5 acres?
- 87,120
- 65,340
- 29,040
- 43,561.5
Show answer and explanation
Answer: B. 65,340
1.5 x 43,560 = 65,340 square feet.
2. A rectangular lot is 120 feet by 180 feet. What is its area?
- 300 linear feet
- 600 square feet
- 2.02 acres
- 21,600 square feet
Show answer and explanation
Answer: D. 21,600 square feet
120 x 180 = 21,600 square feet.
3. A triangle has a 200-foot base and a 120-foot perpendicular height. What is its area?
- 10,000 square feet
- 320 square feet
- 12,000 square feet
- 24,000 square feet
Show answer and explanation
Answer: C. 12,000 square feet
200 x 120 / 2 = 12,000 square feet.
4. What is the nominal acreage of the NW 1/4 of the SE 1/4 of a standard section?
- 80 acres
- 40 acres
- 20 acres
- 160 acres
Show answer and explanation
Answer: B. 40 acres
640 x 1/4 x 1/4 = 40 nominal acres.
5. A rectangle contains 18,000 square feet and is 100 feet wide. What is its length?
- 180 feet
- 18 feet
- 1,800 feet
- 280 feet
Show answer and explanation
Answer: A. 180 feet
18,000 square feet / 100 feet = 180 feet.
6. A rectangular lot measures 60 yards by 90 feet. How many square feet does it contain?
- 5,400
- 48,600
- 1,800
- 16,200
Show answer and explanation
Answer: D. 16,200
Convert 60 yards to 180 feet, then multiply 180 x 90 = 16,200 square feet. The product 1,800 obtained in square yards must be multiplied by nine.
7. A trapezoid has parallel sides of 150 and 250 feet, separated by a perpendicular height of 180 feet. What is its area?
- 72,000 square feet
- 27,000 square feet
- 36,000 square feet
- 45,000 square feet
Show answer and explanation
Answer: C. 36,000 square feet
(150 + 250) / 2 x 180 = 36,000 square feet. Use both parallel sides and their perpendicular separation.
8. A circular parcel has a diameter of 100 feet. Use pi = 3.14. What is its area?
- 15,700 square feet
- 7,850 square feet
- 31,400 square feet
- 314 square feet
Show answer and explanation
Answer: B. 7,850 square feet
Radius is 100 / 2 = 50 feet. Area is 3.14 x 50 x 50 = 7,850 square feet. Using the diameter as radius quadruples the result.
9. A rectangular floor is 18 feet by 15 feet. Materials cost $36 per square yard, with no waste or other charges. What is the stated price?
- $1,080
- $30
- $3,240
- $9,720
Show answer and explanation
Answer: A. $1,080
Area is 270 square feet, or 270 / 9 = 30 square yards. Multiply 30 by $36 to obtain $1,080.
10. An L-shaped lot occupies a 200-by-150-foot rectangle except for a missing 50-by-40-foot corner. What is its area?
- 30,000 square feet
- 2,000 square feet
- 32,000 square feet
- 28,000 square feet
Show answer and explanation
Answer: D. 28,000 square feet
Subtract the actual cutout once: 200 x 150 - 50 x 40 = 30,000 - 2,000 = 28,000 square feet.
11. Which statement about a one-acre parcel is correct?
- Its frontage must be 660 feet
- Its area is 43,560 square feet, but its shape and side lengths are not fixed
- Every side must be about 208.71 feet
- It always has a shorter perimeter than every parcel of the same area
Show answer and explanation
Answer: B. Its area is 43,560 square feet, but its shape and side lengths are not fixed
An acre is an area unit. A square acre has sides of about 208.71 feet, but rectangles and many other shapes can contain the same area.
12. A question gives the actual surveyed area of an irregular government section as 620 acres. May you replace that fact with the nominal 640-acre section figure?
- Yes, whenever the result is easier to calculate
- No; use the actual supplied acreage rather than the nominal shortcut
- Yes, if the section is in Illinois
- Yes, all sections must contain exactly 640 acres
Show answer and explanation
Answer: B. No; use the actual supplied acreage rather than the nominal shortcut
The standard section model is nominal. The question's actual area controls its arithmetic. Also, a legally described half of an irregular section is not automatically half its total acreage.
Common questions
How many square feet are in one acre?
One acre contains exactly 43,560 square feet in the ordinary exam conversion. Multiply acres by 43,560 to obtain square feet. Divide square feet by 43,560 to obtain acres.
How do you calculate the area of a rectangular lot?
Multiply length by width after converting both dimensions to compatible units. A lot measuring 330 feet by 264 feet contains 87,120 square feet, or exactly 2 acres after dividing by 43,560.
How do you calculate the area of a triangular lot?
Multiply the base by the perpendicular height and divide by 2. A 300-foot base with a 220-foot perpendicular height contains 33,000 square feet. A sloping boundary is not the height unless it meets the base at a right angle.
How do you calculate the area of a circular parcel?
Use pi times radius squared. If the diameter is given, divide by 2 to find radius first. Follow the value of pi and rounding instruction in the question; if none is supplied, retain calculator precision until the final answer.
How many square feet are in a square yard?
One square yard contains 9 square feet because one yard equals 3 feet in each of two dimensions. The area conversion is 3 times 3, not merely 3.
How many acres are in a square mile and section?
One square mile contains 640 acres. A standard government section is nominally one square mile, so exam questions commonly use 640 nominal acres. Actual surveyed sections can include irregularities and corrections. A legally described half of an irregular section is not automatically half its total acreage; use the parcel-specific facts rather than blindly dividing the whole-section area.
How many acres are in a survey township?
A nominal survey township is 6 miles by 6 miles, contains 36 square miles or sections, and therefore contains 23,040 nominal acres. Do not confuse a survey township with a civil township or assume every surveyed township has exact nominal acreage.
How do you calculate the area of an irregular lot?
Divide the lot into shapes whose formulas you know, calculate each part in one unit, and add the included pieces. If the drawing contains a cutout, calculate the larger enclosing shape and subtract the missing piece. Do not add perimeter lengths to obtain area.
What is the difference between frontage and area?
Frontage is one linear dimension along a street, water body, or other boundary. Area measures a two-dimensional surface. Frontage alone cannot determine acreage without depth or enough other geometry.
Are these official Illinois broker exam questions?
No. These are original teaching questions supporting the June 24, 2026 national broker outline. The whole Property Ownership domain has 10 of 100 scored national questions; PSI does not publish a separate quota for this page. A practice score is not a validated prediction that you will pass.
What is the formula for a trapezoidal lot?
Add the two parallel side lengths, divide by two, then multiply by their perpendicular separation: area = (base 1 + base 2) / 2 x height. Do not substitute a sloping boundary for the perpendicular height.
How long is each side of one acre?
There is no fixed side length unless the shape is specified. A square acre has a side of square root of 43,560, or about 208.71 feet. A 66-by-660-foot rectangle also contains one acre. Frontage alone does not tell you the acreage.
How do chains, rods and acres connect?
One chain is 66 feet or four rods; one rod is 16.5 feet. A square chain is 66 x 66 = 4,356 square feet, and ten square chains make one acre. One mile is 80 chains or 320 rods. Keep linear chains separate from square chains.
Did retirement of the U.S. survey foot change the exam's 43,560 conversion?
No. NIST's 2026 tables still give one acre as 43,560 square feet. General use of the U.S. survey foot was retired after December 31, 2022, with historic and legacy applications excepted. The international foot and legacy survey foot have different meter conversions. Do not silently mix those definitions in historical survey work.
Can I infer a missing dimension from the acreage?
For a stated rectangle, divide area by the known side: 21,780 square feet / 110 feet = 198 feet. For a stated square, take the square root of its area. Without a defined shape and enough measurements, there may be many possible dimensions.
Is gross floor area the same as living, usable or rentable area?
Do not assume so. Use the definition and included spaces the question supplies. Multiplying outside dimensions does not by itself establish which rooms, garages, shared spaces or other areas belong in a particular reported metric. This page does not supply a building-measurement standard.
How should I check the answer before moving on?
Write the shape formula, the converted dimensions and the final unit. Reverse a missing-dimension calculation by multiplying the result back. For pricing, make sure the area or frontage unit matches the quoted rate. Keep intermediate precision and follow the stated final rounding instruction.
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Official sources and scope
Checked October 25, 2026. The relevant 2026 NIST conversion entries and publisher geometry sections were read. The earlier verified BLM 2017 and January 2026 Illinois DOR readings are reused for the nominal survey model and its limits. These sources support the identified facts; this is not a survey, a building-measurement standard or a review of every chapter in the cited publications.
- PSI Illinois booklet, June 24, 2026: measuring structures and land
- OpenStax Prealgebra 2e 9.4: rectangles, triangles and trapezoids
- OpenStax Prealgebra 2e 9.5: circles and composite figures
- NIST Handbook 44, 2026 Appendix C: length and area conversions
- NIST revised unit conversion factors: current and legacy survey units
- BLM 2017 Specifications for Descriptions of Land, printed 8-11: nominal units, description order, lots and government-measurement halves
- Illinois DOR PTAX-1-M, January 2026: measurements, rectangular survey, political townships and aliquot exercises
This is exam preparation, not advice for a specific transaction. State law and the applicable facts control. Our editorial and corrections process explains how to report an issue.