Area, Perimeter and Cubic Feet for the New York Exam

The arithmetic becomes reliable when the unit matches the question. Feet, square feet and cubic feet are different quantities, even when they use the same dimensions.

What formulas should New York students know?

Use π ≈ 3.1416 unless the problem or calculator directs you differently. Round at the end rather than shortening intermediate values too early.

Official source map

The New York State Department of State 77-hour curriculum lists calculating area and length under Subject 10, Real Estate Mathematics. It expressly includes square feet, irregular lot size, perimeter, acre, hectare and price per square foot.

The Department of State salesperson page says the multiple-choice examination is based on that curriculum and allows 1 1/2 hours after instructions. The Department does not publish an official count or weighting for measurement questions.

The National Institute of Standards and Technology explains that a computed area uses the square of the input unit and a computed volume uses the cube of the input unit in its circumference, area and volume guide. NIST's 2026 general tables of units confirm that one square foot equals 144 square inches and one cubic foot equals 1,728 cubic inches.

The U.S. Department of Agriculture Natural Resources Conservation Service provides official surveying formulas for rectangles, circles and triangles. These geometry sources support the calculations below. They do not establish how a particular property must be measured, advertised, appraised or taxed.

What is the difference between length, area and volume?

Length is one-dimensional

Length measures a line or distance. Examples include:

• a 14-foot wall
• a 100-foot lot boundary
• a 72-foot fence segment

The answer stays in feet, inches, yards or another linear unit.

Area is two-dimensional

Area measures a flat surface. Multiplying feet by feet produces square feet:

12 ft × 10 ft = 120 ft²

Area can describe a room floor, parcel segment, wall face or roof plane.

Volume is three-dimensional

Volume includes height or depth:

12 ft × 10 ft × 8 ft = 960 ft³

Volume can describe the space inside a rectangular room, the capacity of a rectangular container or the amount of material in a rectangular solid.

A square foot is not a foot, and a cubic foot is not a square foot. Label the unit beside every answer.

How do you calculate the area of a rectangle?

Use:

Area = Length × Width

Example 1: Rectangular room

A room is 18 feet long and 14 feet wide.

Area = 18 ft × 14 ft = 252 ft²

The floor area is 252 square feet.

If carpet costs $4.50 per square foot in a simplified problem:

252 × $4.50 = $1,134

The stated carpet cost is $1,134 before waste, labor, tax or other amounts unless the problem includes them.

How do you find a missing rectangle dimension?

Rearrange the area formula:

Missing dimension = Area ÷ Known dimension

Example 2: Find width

A rectangular office contains 1,680 square feet and is 48 feet long.

Width = 1,680 ft² ÷ 48 ft = 35 ft

The width is 35 feet.

Notice how square feet divided by feet leaves feet. The units help confirm the setup.

How do you calculate square area?

All four sides of a square have equal length.

Area = Side²

Example 3

A square courtyard has 22-foot sides.

Area = 22 ft × 22 ft = 484 ft²

The courtyard area is 484 square feet.

Do not multiply the side by four when the question asks for area. Four times the side finds perimeter.

How do you calculate triangle area?

Use:

Area = Base × Perpendicular height ÷ 2

The height must meet the base at a right angle. A slanted side is not automatically the height.

Example 4: Triangular parcel segment

A triangular segment has a 90-foot base and a perpendicular height of 64 feet.

Area = 90 ft × 64 ft ÷ 2

Area = 2,880 ft²

The segment contains 2,880 square feet.

The division by two matters because a triangle with the same base and perpendicular height occupies half the area of the related rectangle.

How do you calculate parallelogram area?

Use:

Area = Base × Perpendicular height

Do not substitute the slanted side for the perpendicular height.

Example 5

A parallelogram has a 75-foot base and a perpendicular height of 40 feet.

Area = 75 ft × 40 ft = 3,000 ft²

The area is 3,000 square feet.

How do you calculate trapezoid area?

A trapezoid has one pair of parallel sides. Use:

Area = (Parallel side 1 + Parallel side 2) × Perpendicular height ÷ 2

Another way to read the formula is: find the average length of the two parallel sides, then multiply by the height.

Example 6

A trapezoidal parcel section has parallel sides of 120 feet and 180 feet. The perpendicular distance between them is 70 feet.

Average parallel side = (120 + 180) ÷ 2 = 150 ft

Area = 150 ft × 70 ft = 10,500 ft²

The area is 10,500 square feet.

How do you calculate circle area?

Use:

Area = π × Radius²

The radius runs from the center to the edge. The diameter runs across the entire circle through the center, so:

Radius = Diameter ÷ 2

Example 7: Diameter supplied

A circular feature has a diameter of 30 feet.

Radius = 30 ft ÷ 2 = 15 ft

Area = 3.1416 × 15²

Area = 706.86 ft²

The area is approximately 706.86 square feet.

Do not square the diameter in the radius formula. Convert diameter to radius first.

What is perimeter?

Perimeter is the total distance around the outside boundary of a two-dimensional shape. It uses linear units.

For a rectangle:

Perimeter = 2 × Length + 2 × Width

Or:

Perimeter = 2 × (Length + Width)

Example 8: Rectangle perimeter

A rectangular yard measures 80 feet by 55 feet.

Perimeter = 2 × 80 ft + 2 × 55 ft

Perimeter = 160 ft + 110 ft = 270 ft

The perimeter is 270 feet.

Multiplying 80 by 55 would find area, not the amount of boundary fencing.

How do you calculate an irregular perimeter?

Add every outside edge once. Do not include an interior dividing line unless the question treats it as part of the boundary being measured.

Example 9: Five-sided outline

An irregular outline has outside sides measuring 35, 42, 28, 51 and 44 feet.

Perimeter = 35 + 42 + 28 + 51 + 44 = 200 ft

The perimeter is 200 feet.

The shape does not need a special perimeter formula when every outside side is given.

How do you calculate the circumference of a circle?

Circumference is a circle's perimeter.

Use either:

Circumference = 2 × π × Radius

Or:

Circumference = π × Diameter

Example 10

A circular garden has a 24-foot diameter.

Circumference = 3.1416 × 24 ft = 75.3984 ft

Rounded to the nearest tenth, the circumference is 75.4 feet.

Circle area would be stated in square feet. Circumference is stated in feet.

How do you calculate cubic feet?

For a rectangular solid:

Volume = Length × Width × Height

Example 11: Room volume

A rectangular room measures 16 feet by 12 feet and has a 9-foot ceiling.

Volume = 16 ft × 12 ft × 9 ft

Volume = 1,728 ft³

The room contains 1,728 cubic feet of space under the simplified rectangular model.

Example 12: Missing height

A rectangular space has a volume of 4,800 cubic feet. Its floor measures 30 feet by 20 feet.

First find floor area:

30 ft × 20 ft = 600 ft²

Then divide volume by floor area:

Height = 4,800 ft³ ÷ 600 ft² = 8 ft

The height is 8 feet.

How do you convert square and cubic units?

Linear conversion factors must be squared for area and cubed for volume.

Since one foot equals 12 inches:

1 ft² = 12 in × 12 in = 144 in²

1 ft³ = 12 in × 12 in × 12 in = 1,728 in³

Since one yard equals 3 feet:

1 yd² = 3 ft × 3 ft = 9 ft²

1 yd³ = 3 ft × 3 ft × 3 ft = 27 ft³

NIST's 2026 tables confirm these relationships. Avoid using 12 as the square-foot conversion or 3 as the square-yard conversion.

Example 13: Square yards to square feet

A surface contains 46 square yards.

46 yd² × 9 = 414 ft²

The area is 414 square feet.

Example 14: Cubic yards to cubic feet

A volume is 8 cubic yards.

8 yd³ × 27 = 216 ft³

The volume is 216 cubic feet.

How do you solve an L-shaped area?

An L shape can be divided into two non-overlapping rectangles or treated as one large rectangle minus a missing rectangle. Both methods should agree.

Example 15: Add two rectangles

An L-shaped floor can be divided into:

• Rectangle A: 30 feet by 20 feet
• Rectangle B: 12 feet by 10 feet

Rectangle A = 30 × 20 = 600 ft²

Rectangle B = 12 × 10 = 120 ft²

Total area = 600 + 120 = 720 ft²

The L-shaped floor contains 720 square feet.

The two rectangles must not overlap. If they overlap, the shared portion would be counted twice.

How do you use the subtract method for an irregular shape?

Draw or imagine a complete outer rectangle, then subtract the missing cutout.

Example 16: Outer rectangle minus cutout

An outer rectangle is 40 feet by 32 feet. A rectangular 14-foot by 8-foot section is missing from one corner.

Outer area = 40 × 32 = 1,280 ft²

Cutout area = 14 × 8 = 112 ft²

Remaining area = 1,280 - 112 = 1,168 ft²

The irregular shape contains 1,168 square feet.

Use subtraction only when the larger rectangle truly contains the shape and the stated cutout accounts for the missing portion.

How do you divide a mixed irregular shape?

The U.S. Department of Agriculture describes a practical method: segment an irregular area into regular shapes and add the segment areas. In exam-style problems, those shapes are often rectangles and triangles.

Example 17: Rectangle plus triangle

A parcel diagram can be divided into:

• a rectangle measuring 100 feet by 60 feet
• a triangle with a 40-foot base and a 60-foot perpendicular height

Calculate each part:

Rectangle = 100 × 60 = 6,000 ft²

Triangle = 40 × 60 ÷ 2 = 1,200 ft²

Total = 6,000 + 1,200 = 7,200 ft²

The combined area is 7,200 square feet.

Do not add dimensions before deciding how the shapes fit. Calculate each non-overlapping section, preserve its square-foot unit, then total the areas.

How should you handle a drawing that is not to scale?

Use the printed dimensions, not the apparent proportions. A short-looking side can have a larger stated measurement than a long-looking side. “Not to scale” means the visual is a map of relationships, not a ruler.

Before calculating:

1. mark every known dimension
2. identify any opposite sides with equal lengths
3. derive missing segments only from valid relationships
4. divide the figure into familiar shapes
5. verify that the parts cover the entire area without gaps or overlap

If a necessary dimension cannot be derived from the facts, the problem does not provide enough information.

Is calculated floor area the same as official living area?

Not necessarily. Geometry can calculate the dimensions supplied in an educational problem. Actual reporting of gross living area, rentable area, usable area or building area can depend on definitions, measurement standards, inclusions, exclusions and professional judgment.

Do not turn a room-by-room practice sum into a representation about a real property without applying the relevant standard and verifying the measurements. This lesson teaches mathematical relationships, not an appraisal or property measurement certification.

What are the most common measurement mistakes?

Using perimeter when the question asks for area

Fencing usually suggests linear boundary length. Flooring usually suggests surface area. Read the requested unit.

Forgetting to square the unit

Length times width produces square units, not linear feet.

Forgetting the third dimension

Volume requires length, width and height. Floor area alone is not cubic volume.

Multiplying triangle base by height without dividing by two

A triangle occupies half the related rectangle or parallelogram with the same base and perpendicular height.

Using diameter as radius

Radius is half the diameter. Circle area uses the radius squared.

Adding an interior line to perimeter

Perimeter follows the outside boundary unless the problem expressly includes another segment.

Double-counting overlapping rectangles

Composite pieces must be non-overlapping, or the overlap must be subtracted once.

Converting square or cubic units with a linear factor

One square yard is 9 square feet, and one cubic yard is 27 cubic feet.

Rounding too early

Keep the calculator's full intermediate value, especially with circles, then round as requested.

What is a reliable exam-day workflow?

Use this sequence:

1. Circle the requested quantity and unit.
2. Sketch or label the shape from the stated dimensions.
3. Convert all dimensions to the same linear unit.
4. Choose the formula for each shape.
5. Calculate non-overlapping parts.
6. Add or subtract the parts as the diagram requires.
7. Label the final answer with linear, square or cubic units.
8. Estimate whether the result fits inside the outer dimensions.

For example, an irregular shape inside a 40-by-30-foot outer rectangle cannot have more than 1,200 square feet unless part of it extends outside that boundary.

Use the New York real estate math formula map to connect measurement formulas with percentages, finance, taxes and prorations.

Review the complete Real Estate Mathematics study guide when you want the formulas placed beside the other Subject 10 objectives.

Can you solve these original practice questions?

Practice 1

A rectangular room measures 17 feet by 13 feet. What is its floor area?

A. 60 square feet
B. 221 square feet
C. 442 square feet
D. 2,652 square feet

Answer: B. 17 × 13 = 221 square feet.

Practice 2

A rectangular lot segment is 110 feet long and 70 feet wide. What is its perimeter?

A. 180 feet
B. 360 feet
C. 7,700 feet
D. 15,400 feet

Answer: B. 2 × 110 + 2 × 70 = 360 feet.

Practice 3

A triangular section has an 84-foot base and a 50-foot perpendicular height. What is its area?

A. 1,050 square feet
B. 2,100 square feet
C. 4,200 square feet
D. 8,400 square feet

Answer: B. 84 × 50 ÷ 2 = 2,100 square feet.

Practice 4

A circular patio has a 20-foot diameter. Using 3.1416 for π, what is its approximate area?

A. 62.83 square feet
B. 125.66 square feet
C. 314.16 square feet
D. 1,256.64 square feet

Answer: C. The radius is 10 feet. 3.1416 × 10² = 314.16 square feet.

Practice 5

A rectangular storage space measures 25 feet by 18 feet by 10 feet. What is its volume?

A. 450 cubic feet
B. 900 cubic feet
C. 4,500 cubic feet
D. 45,000 cubic feet

Answer: C. 25 × 18 × 10 = 4,500 cubic feet.

Practice 6

An irregular floor consists of one 24-by-20-foot rectangle and one non-overlapping 10-by-8-foot rectangle. What is the total area?

A. 480 square feet
B. 520 square feet
C. 560 square feet
D. 800 square feet

Answer: C. The parts are 24 × 20 = 480 and 10 × 8 = 80. Total area is 480 + 80 = 560 square feet.

Practice 7

How many cubic feet are in 12 cubic yards?

A. 36 cubic feet
B. 108 cubic feet
C. 144 cubic feet
D. 324 cubic feet

Answer: D. One cubic yard is 27 cubic feet. 12 × 27 = 324 cubic feet.

Practice 8

Which unit correctly states the answer to a length-times-width calculation when both dimensions are in feet?

A. Feet
B. Square feet
C. Cubic feet
D. Feet per square foot

Answer: B. Feet multiplied by feet produces square feet.

Frequently asked questions

What is the formula for square footage?

For a rectangle, multiply length by width. State the answer in square feet when both dimensions are in feet.

What is the difference between area and perimeter?

Area measures the surface inside a boundary in square units. Perimeter measures the distance around the boundary in linear units.

How do I calculate an irregular lot area?

Divide the diagram into familiar non-overlapping shapes, calculate each area, then add the parts. Alternatively, subtract a clearly defined cutout from a larger enclosing shape.

What is the formula for cubic feet?

For a rectangular solid, multiply length by width by height. Use feet for all three dimensions so the answer is in cubic feet.

How many square feet are in one square yard?

One square yard equals 9 square feet because a yard is 3 feet and 3 × 3 = 9.

How many cubic feet are in one cubic yard?

One cubic yard equals 27 cubic feet because 3 × 3 × 3 = 27.

Does the New York exam publish a measurement-question count?

No official topic count is published. The Department of State curriculum includes area, length, irregular lot size and perimeter, but it does not state a subject weighting.

Sources and verification notes

This article was checked against official materials available on August 27, 2026. The practice diagrams are described in words so each problem supplies the dimensions needed to solve it. The examples are original and do not reproduce state examination questions.

• New York State Department of State, Real Estate Salesperson 77-Hour Curriculum
• New York State Department of State, Real Estate Salesperson
• National Institute of Standards and Technology, Circumference, Area and Volume
• National Institute of Standards and Technology, Handbook 44, 2026 Appendix C
• National Institute of Standards and Technology, Revised Unit Conversion Factors
• U.S. Department of Agriculture Natural Resources Conservation Service, Construction Surveying and Quantity Computations

Use these calculations for education and exam preparation. For an actual property, use verified measurements and the definition or professional standard required for the intended purpose.