- Official section
- National XI.H: Area Calculations
- Broker weight
- A named skill within 7% of the national portion
- Expected scored items
- The current PSI broker outline assigns about 7 of 100 scored national items to Real Estate Math
Land and building math guide
Area calculations with units that cancel cleanly
Area math rewards labels. Write every dimension with its unit, calculate the shape before converting, and put a square on the answer unit. That small habit prevents the classic errors of treating frontage as area, using 3 instead of 9 for square yards, or multiplying by 43,560 in the wrong direction.
Last updated: August 1, 2026
What calculation do you need to make?
Short answer: Rectangle area equals length times width. Square area equals side squared. Triangle area equals one-half base times perpendicular height. Circle area equals pi times radius squared. Trapezoid area equals one-half times the sum of the parallel sides times perpendicular height. One acre contains 43,560 square feet or 4,840 square yards. One square yard contains 9 square feet. One chain is 66 feet, one rod is 16.5 feet, and one mile is 80 chains or 320 rods. One square mile or nominal section contains 640 acres, and a nominal survey township contains 36 sections or 23,040 acres. Convert all dimensions to compatible linear units before calculating, keep square units separate from linear units, divide irregular parcels into simpler shapes, and round only when instructed.
The PSI outline expressly names area calculations. Exam diagrams are commonly simplified and drawn for arithmetic rather than survey control. Actual parcel area follows controlling boundaries, monuments, senior rights, curves, record evidence, applicable law, and professional survey work; it can differ from nominal PLSS acreage or dimensions multiplied from a sketch. NIST retired general use of the U.S. survey foot after December 31, 2022 except for historic and legacy applications, but historical records may retain that unit. When a historic record expressly identifies a legacy unit, use its specified conversion rather than silently substituting a modern unit or a rounded field estimate. That distinction can matter in archived plats, deeds, and surveys. This guide teaches exam geometry, not surveying, architecture, engineering, or legal-description interpretation. Sources were checked through August 1, 2026.
How do you solve area and acreage questions?
- Write the requested answer unit before copying any dimensions from the problem.
- Sketch or identify the shape and mark which dimensions are length, width, base, perpendicular height, radius, or diameter.
- Convert all linear dimensions into one compatible unit before multiplying them.
- Apply the rectangle, square, triangle, circle, trapezoid, addition, or subtraction formula and label the result in square units.
- Convert square feet to acres by dividing by 43,560 or acres to square feet by multiplying by 43,560.
- For section fractions, begin with 640 nominal acres and apply each fraction from the full section inward.
- Keep full precision through intermediate geometry and convert or round only at the requested stage.
- Estimate the size, reverse-check a missing dimension, and confirm that the final label is linear, square, acreage, or price as requested.
- Problem
- Rectangle
- Formula or conversion
- Length x width
- Exam safeguard
- Use compatible units
- Problem
- Square
- Formula or conversion
- Side x side
- Exam safeguard
- Area unit is squared
- Problem
- Triangle
- Formula or conversion
- Base x perpendicular height / 2
- Exam safeguard
- Do not use sloped side
- Problem
- Circle
- Formula or conversion
- Pi x radius x radius
- Exam safeguard
- Diameter / 2 first
- Problem
- Trapezoid
- Formula or conversion
- (Parallel side 1 + side 2) / 2 x height
- Exam safeguard
- Height is perpendicular
- Problem
- Square feet to acres
- Formula or conversion
- Square feet / 43,560
- Exam safeguard
- Moving to larger unit
- Problem
- Acres to square feet
- Formula or conversion
- Acres x 43,560
- Exam safeguard
- Moving to smaller unit
- Problem
- Square yards to square feet
- Formula or conversion
- Square yards x 9
- Exam safeguard
- Square the factor 3
- Problem
- 1 chain
- Formula or conversion
- 66 feet or 4 rods
- Exam safeguard
- Linear survey unit
- Problem
- 1 rod
- Formula or conversion
- 16.5 feet
- Exam safeguard
- 320 rods per mile
- Problem
- Square mile or section
- Formula or conversion
- 640 nominal acres
- Exam safeguard
- Not one linear mile
- Problem
- Survey township
- Formula or conversion
- 36 sections or 23,040 nominal acres
- Exam safeguard
- Actual survey can vary
Can you follow the calculation from facts to answer?
Convert a rectangular lot to acres
Scenario: 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
Scenario: 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
Scenario: 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
Scenario: 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
Scenario: 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
Scenario: 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.
Which math errors cost the most points?
- Trap
- Add length and width to find area.
- Correction
- Adding sides helps find perimeter; rectangle area requires multiplication.
- Trap
- Report a two-dimensional result in linear feet.
- Correction
- Area must be labeled in square feet, square yards, acres, or another area unit.
- Trap
- Convert square yards to square feet with a factor of 3.
- Correction
- One square yard equals 9 square feet because both dimensions convert.
- Trap
- Multiply square feet by 43,560 to get acres.
- Correction
- Divide square feet by 43,560; multiply only when converting acres to square feet.
- Trap
- Use the diameter as radius in circle area.
- Correction
- Divide diameter by two before squaring the radius.
- Trap
- Use a triangle's sloped edge as its height.
- Correction
- Height must be perpendicular to the selected base.
- Trap
- Add overlapping composite shapes without adjustment.
- Correction
- Choose nonoverlapping pieces or subtract the overlap once.
- Trap
- Mix feet and yards in the same multiplication.
- Correction
- Convert both dimensions to one linear unit before calculating area.
- Trap
- Call one linear mile 640 acres.
- Correction
- One square mile contains 640 acres; one mile alone measures length.
- Trap
- Assume every surveyed section contains exactly 640 acres.
- Correction
- The section figure is nominal; use actual acreage when the problem supplies it.
- Trap
- Round the first shape before adding an irregular parcel.
- Correction
- Retain precision through every component and round the final requested result.
- Trap
- Treat a sketch calculation as a controlling survey.
- Correction
- Exam geometry is simplified; actual boundaries and area require controlling evidence and professional survey work.
Can you solve these original problems?
These questions are original study items aligned to the published outline. They are not copied, recalled, or predicted PSI questions.
1. How many square feet are in 1.5 acres?
- 65,340
- 29,040
- 43,561.5
- 87,120
Show answer and explanation
Answer: 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?
- 21,600 square feet
- 600 square feet
- 300 linear feet
- 2.02 acres
Show answer and explanation
Answer: 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?
- 12,000 square feet
- 24,000 square feet
- 320 square feet
- 10,000 square feet
Show answer and explanation
Answer: 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?
- 40 acres
- 20 acres
- 80 acres
- 160 acres
Show answer and explanation
Answer: 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: 180 feet
18,000 square feet / 100 feet = 180 feet.
Which numbers and formulas are easy to confuse?
- Terms
- Linear foot vs. square foot
- Difference
- A linear foot measures one dimension. A square foot measures a surface one foot by one foot.
- Question cue
- Boundary length versus surface area.
- Terms
- Frontage vs. area
- Difference
- Frontage is a boundary length. Area also requires depth or enough other geometry to measure a surface.
- Question cue
- One dimension versus two-dimensional result.
- Terms
- Perimeter vs. area
- Difference
- Perimeter adds boundary lengths. Area measures the enclosed surface through a shape formula.
- Question cue
- Distance around versus space inside.
- Terms
- Square yard vs. square foot
- Difference
- A square yard is 3 feet by 3 feet and therefore equals 9 square feet.
- Question cue
- Square the linear conversion.
- Terms
- Mile vs. square mile
- Difference
- A mile is 5,280 linear feet. A square mile is one mile by one mile and contains 640 acres.
- Question cue
- Length versus large area.
- Terms
- Radius vs. diameter
- Difference
- Radius runs from center to edge. Diameter crosses the circle through its center and equals two radii.
- Question cue
- Half-width versus full-width.
- Terms
- Triangle side vs. perpendicular height
- Difference
- A boundary side can slope. Height is the perpendicular distance from the selected base to the opposite vertex.
- Question cue
- Edge length versus right-angle distance.
- Terms
- Section vs. township
- Difference
- A nominal section contains 640 acres. A nominal survey township contains 36 sections or 23,040 acres.
- Question cue
- One grid square versus 36-square grid.
- Terms
- Nominal acreage vs. surveyed acreage
- Difference
- Nominal acreage follows theoretical grid or shape math. Surveyed acreage reflects controlling evidence and actual professional measurement.
- Question cue
- Exam grid value versus field result.
- Terms
- Gross area vs. net or usable area
- Difference
- Gross area follows its stated outer or total definition. Net or usable area excludes spaces under the applicable measurement definition.
- Question cue
- Total envelope versus defined usable portion.
What does the outline expect you to calculate?
- Topic
- Linear versus square units
- What to know
- Inch, foot, yard, mile, frontage, depth, perimeter, linear foot, square foot, square yard, acre, square mile, dimension, and exponent
- Best exam move
- Use one dimension for length and two multiplied dimensions for area; label the result with square units.
- Topic
- Compatible dimensions
- What to know
- Feet, yards, inches, miles, conversion, mixed units, length, width, base, height, radius, denominator, and unit cancellation
- Best exam move
- Convert all dimensions to one linear unit before applying the shape formula.
- Topic
- Rectangle formula
- What to know
- Length, width, frontage, depth, opposite sides, right angle, square feet, acres, floor area, lot area, gross area, and multiplication
- Best exam move
- Multiply perpendicular length and width, then convert the square-unit result if needed.
- Topic
- Square formula
- What to know
- Equal sides, side squared, square lot, floor tile, area, perimeter, missing side, square root, foot, yard, and acre
- Best exam move
- Multiply the side by itself; use square root only when area is known and side is missing.
- Topic
- Triangle formula
- What to know
- One-half, base, perpendicular height, right angle, sloping side, diagonal, triangular lot, square feet, subdivision, and multiplication
- Best exam move
- Use perpendicular height rather than an unlabeled sloped boundary.
- Topic
- Circle formula
- What to know
- Pi, radius, radius squared, diameter, circumference, circular parcel, cul-de-sac, round building, calculator pi, 3.14, and rounding
- Best exam move
- Divide diameter by two before squaring and do not substitute circumference for area.
- Topic
- Trapezoid formula
- What to know
- Parallel sides, bases, sum, one-half, perpendicular height, irregular parcel, frontage, rear line, square feet, and diagram
- Best exam move
- Add the parallel sides, divide their sum by two, and multiply by perpendicular height.
- Topic
- Composite shape addition
- What to know
- L-shape, rectangle, triangle, wing, addition, shared boundary, no overlap, floor plan, parcel sketch, subtotal, and total area
- Best exam move
- Split the figure into nonoverlapping shapes, calculate each subtotal, and add once.
- Topic
- Cutout subtraction
- What to know
- Enclosing rectangle, missing corner, courtyard, easement area, excluded parcel, notch, larger shape, smaller shape, subtraction, and net area
- Best exam move
- Calculate the complete outer shape and subtract the clearly excluded area.
- Topic
- Missing dimension
- What to know
- Known area, known length, unknown width, division, rectangle, frontage, depth, height, base, square side, square root, and unit label
- Best exam move
- Rearrange the shape formula and divide area by the known perpendicular dimension.
- Topic
- Square feet and acres
- What to know
- 43,560 square feet, acre, multiply, divide, decimal acre, square-foot result, exact acre, rounding, lot size, and conversion factor
- Best exam move
- Multiply acres to obtain square feet; divide square feet to obtain acres.
- Topic
- Square yards and acres
- What to know
- 9 square feet, 4,840 square yards, acre, yard, foot, length conversion, area conversion, carpet, paving, and dimensional square
- Best exam move
- Use 9, not 3, between square feet and square yards.
- Topic
- Square mile scale
- What to know
- 5,280 feet, one mile, square mile, 27,878,400 square feet, 640 acres, large tract, section, length, and area
- Best exam move
- Distinguish one linear mile from a square mile measuring one mile on each side.
- Topic
- Chains and rods
- What to know
- Chain, 66 feet, rod, pole, perch, 16.5 feet, four rods per chain, 80 chains per mile, 320 rods per mile, square chain, 4,356 square feet, 10 square chains per acre, historic description, and legacy survey unit
- Best exam move
- Convert each linear dimension before calculating area and do not confuse one chain with one square chain.
- Topic
- Section acreage
- What to know
- Section, 640 nominal acres, half section, quarter section, quarter-quarter, aliquot part, nested fraction, government survey, irregular section, and correction
- Best exam move
- Begin with 640 nominal acres and apply every nested fraction once.
- Topic
- Township acreage
- What to know
- Survey township, 6 miles by 6 miles, 36 square miles, 36 sections, 23,040 acres, nominal grid, civil township, range, and correction line
- Best exam move
- Use 36 times 640 for a nominal full-township question and preserve the word nominal.
- Topic
- Floor and usable area
- What to know
- Exterior dimensions, interior dimensions, gross building area, gross living area, rentable area, usable area, common area, stair opening, wall thickness, measurement standard, and definition
- Best exam move
- Use the area definition and included spaces stated in the question rather than assuming every building metric is identical.
- Topic
- Price per area unit
- What to know
- Total price, price per square foot, price per acre, area, multiplication, division, land value, building value, unit rate, comparable, and conversion
- Best exam move
- Match the price unit to the area unit before multiplying or dividing.
- Topic
- Precision and surveyed reality
- What to know
- Exact conversion, decimal acre, calculator precision, pi, rounding, significant digits, nominal acreage, deed acreage, survey acreage, boundary, monument, and professional survey
- Best exam move
- Keep full precision through the math and prefer supplied surveyed acreage over a nominal shortcut.
How should you drill this calculation?
- Session
- Session 1
- Focus
- Master unit labels
- Proof you are ready
- Classify 40 facts and answers as linear, square, cubic, acreage, or price units before calculating.
- Session
- Session 2
- Focus
- Memorize fixed conversions
- Proof you are ready
- Write the acre, square-yard, square-mile, section, township, mile, chain, and rod relationships from memory.
- Session
- Session 3
- Focus
- Solve core shapes
- Proof you are ready
- Complete ten rectangle, ten triangle, ten circle, and ten trapezoid problems with labeled units.
- Session
- Session 4
- Focus
- Decompose irregular shapes
- Proof you are ready
- Split, add, or subtract 20 L-shaped, notched, composite, and excluded-area diagrams.
- Session
- Session 5
- Focus
- Reverse and price area
- Proof you are ready
- Find 15 missing dimensions and 15 price-per-square-foot or price-per-acre results with unit cancellation.
- Session
- Session 6
- Focus
- Complete a mixed area set
- Proof you are ready
- Score at least 90% and justify each answer by shape, dimensions, formula, conversion direction, square unit, precision, and reality check.
Do not count recognition as mastery. Close the notes and explain the rule, apply it to a new fact pattern, and identify why each distractor fails.
Build speed without skipping the setup
From concept to decision
Drill this topic, then review the explanation
Pass Illinois gives you original national and Illinois questions, topic-by-topic study, clear explanations, timed practice, flashcards, progress tracking, and Math Coach. Start free, find the weak distinction, and focus the next session there.
Questions students ask about Area, Acreage, and Square Footage Calculations
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.
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. They are original calculations aligned to the PSI broker outline effective June 24, 2026. The 2026 NIST measurement tables, current federal land-description materials, and Illinois survey standards were checked through August 1, 2026.
Primary sources
- PSI Illinois Candidate Information Booklet dated June 24, 2026
- NIST Handbook 44, current 2026 edition
- NIST Handbook 44 Appendix C, 2026 General Tables of Units of Measurement
- NIST revised survey-unit conversion factors, updated February 3, 2026
- Bureau of Land Management, federal legal-description and PLSS guidance
- 68 Ill. Adm. Code 1270.56, current Illinois land-survey standards
The current official outline controls the tested scope. Statutes, regulations, and official agency materials control when a general study rule and a jurisdiction-specific rule differ.
Editorial status
Checked against primary sources
The Pass Illinois editorial team last checked this guide on August 1, 2026. Every practice question is an original study item, and the source links above let you verify the rules that support the lesson.