Area Calculator

Master these essential area formulas for accurate calculations:

Rectangle and Square:

The rectangle formula is the most commonly used—it applies to most rooms, lots, and construction projects. For squares, simply multiply one side by itself.

Circle and Ellipse:

Where r = radius (half the diameter), a = semi-major axis, b = semi-minor axis. Circles and ellipses appear in pools, patios, and landscaping features.

Triangle:

The height must be perpendicular to the base. For obtuse triangles, the height may fall outside the triangle.

Trapezoid and Parallelogram:

Where a and b are the parallel sides and h is the perpendicular height between them.

Irregular Polygon:

For complex shapes, use the Shoelace formula with coordinates:

A = ½|Σ(xᵢyᵢ₊₁ - xᵢ₊₁yᵢ)|

Or divide into triangles/rectangles and sum the individual areas.

Convert between area units with these precise conversion factors:

Imperial Units:

Metric Units:

Imperial to Metric:

Quick Reference Sizes:

Apply area calculations to common projects:

Home Improvement Projects:

Landscaping Projects:

Land Measurement:

Cost Estimation Examples (2026):

Handle non-standard shapes with these techniques:

Method 1: Divide and Conquer

Break complex shapes into simple rectangles, triangles, and circles:

1. L-Shaped Rooms: Divide into two rectangles

   • Measure each section separately
   • Calculate: Area1 + Area2 = Total
   • Example: (10×12) + (8×6) = 120 + 48 = 168 sq ft

2. T-Shaped Lots: Three rectangles

   • Main section + left wing + right wing
   • Or: large rectangle - corner cutouts

3. Rooms with Alcoves: Main rectangle + alcove area

   • Measure main room and add bump-outs
   • Subtract closets if calculating floor coverage only

Method 2: Subtraction

Calculate a larger simple shape and subtract cutouts:

1. Start with bounding rectangle (length × width of entire space)
2. Calculate areas of cutouts, notches, or excluded sections
3. Subtract: Total = Bounding Area - Cutouts

Example: Room with bay window

• Bounding rectangle: 15 × 20 = 300 sq ft
• Bay window adds: 3 × 4 = 12 sq ft
• Total: 312 sq ft

Method 3: Coordinate Method (Shoelace Formula)

For irregular polygons, plot corners as coordinates:

1. List all vertices in order: (x₁,y₁), (x₂,y₂), ..., (xₙ,yₙ)
2. Apply: A = ½|Σ(xᵢyᵢ₊₁ - xᵢ₊₁yᵢ)|
3. Include final pair: (xₙ,y₁) - (x₁,yₙ)

Example: Triangle with corners at (0,0), (4,0), (2,3)

• A = ½|(0×0 - 4×0) + (4×3 - 2×0) + (2×0 - 0×3)|
• A = ½|0 + 12 + 0| = 6 square units

Method 4: Grid Approximation

For curved or highly irregular shapes:

1. Overlay a grid on the shape
2. Count full squares inside
3. Count partial squares (estimate as ½)
4. Total: Full squares + (½ × Partial squares)

This method works well for organic shapes like ponds, curved garden beds, or irregular lots.

Tips for Accuracy:

• Always measure twice, calculate once
• Use the same unit throughout (don't mix feet and inches)
• Draw a sketch and label all dimensions
• Round up for material ordering, round to nearest for pricing
• Photograph complex spaces for reference

Master circular and curved area calculations:

Circle Fundamentals:

Circle Area Quick Reference:

Partial Circles (Sectors):

For pizza-slice shapes:

• Sector area: A = (θ/360) × πr²
• Where θ = central angle in degrees

Example: Quarter circle (90°) with 10 ft radius

• A = (90/360) × π × 10² = 78.54 sq ft

Semicircle:

• A = ½πr² = πr²/2
• Example: 8 ft diameter semicircle = ½ × π × 4² = 25.13 sq ft

Ellipse (Oval):

• A = π × a × b
• Where a = semi-major axis (half the long diameter)
• And b = semi-minor axis (half the short diameter)

Example: Oval pool 20 ft long × 12 ft wide

• a = 10 ft, b = 6 ft
• A = π × 10 × 6 = 188.50 sq ft

Annulus (Ring/Donut):

Area between two concentric circles:

• A = π(R² - r²)
• Where R = outer radius, r = inner radius

Example: Circular driveway, outer diameter 30 ft, inner diameter 20 ft

• R = 15 ft, r = 10 ft
• A = π(225 - 100) = π × 125 = 392.70 sq ft

Circular Segments:

Arc-shaped area cut by a chord:

• A = r²×arccos((r-h)/r) - (r-h)×√(2rh-h²)
• Where h = height of segment

This applies to circular window shapes, arch tops, and similar architectural features.

Multiple approaches for calculating triangle area:

Method 1: Base × Height (Most Common)

• A = ½ × base × height
• Height must be perpendicular to base
• Works for all triangle types

Method 2: Heron's Formula (Using Three Sides)

When you know all three sides but not the height:

1. Calculate semi-perimeter: s = (a + b + c) / 2
2. Apply: A = √[s(s-a)(s-b)(s-c)]

Example: Triangle with sides 7, 8, and 9 units

• s = (7 + 8 + 9) / 2 = 12
• A = √[12 × 5 × 4 × 3] = √720 = 26.83 sq units

Method 3: Coordinate Formula

For triangles plotted on a coordinate plane:

• A = ½|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|

Example: Vertices at (1,1), (4,5), (7,2)

• A = ½|1(5-2) + 4(2-1) + 7(1-5)|
• A = ½|3 + 4 - 28| = ½ × 21 = 10.5 sq units

Method 4: Two Sides and Included Angle

When you know two sides and the angle between them:

• A = ½ × a × b × sin(C)

Example: Sides 10 and 12 with 30° angle between

• A = ½ × 10 × 12 × sin(30°) = ½ × 10 × 12 × 0.5 = 30 sq units

Equilateral Triangle Shortcut:

For equilateral triangles (all sides equal):

• A = (√3/4) × s²
• Where s = side length

Example: Equilateral with 10 ft sides

• A = (√3/4) × 100 = 43.30 sq ft

Right Triangle Shortcut:

For right triangles:

• A = ½ × leg₁ × leg₂
• The legs are the two sides forming the right angle

Example: Right triangle with legs 6 and 8

• A = ½ × 6 × 8 = 24 sq units

Avoid these frequent errors that lead to costly miscalculations:

Mistake 1: Confusing Perimeter with Area

• Perimeter: Distance around (linear feet)
• Area: Space inside (square feet)
• A 10×10 room has 40 ft perimeter but 100 sq ft area

Consequence: Ordering 40 sq ft of flooring for a 100 sq ft room.

Mistake 2: Forgetting to Square Units

• Doubling dimensions quadruples area
• 10×10 = 100 sq ft, but 20×20 = 400 sq ft (not 200)

Consequence: Severely underestimating materials for scaled-up projects.

Mistake 3: Mixing Measurement Units

• 12 feet × 18 inches ≠ 216 (that's mixing units)
• Correct: 12 ft × 1.5 ft = 18 sq ft

Consequence: Off by a factor of 12 when mixing feet and inches.

Mistake 4: Using Diameter Instead of Radius

• Circle area uses radius (r), not diameter
• A = πr², NOT πd²
• A 10 ft diameter circle: r = 5, A = π × 25 = 78.5 sq ft

Consequence: Calculating 4× the actual area if using diameter as radius.

Mistake 5: Forgetting Height Must Be Perpendicular

• Triangle/trapezoid height is the perpendicular distance
• Slant height ≠ perpendicular height

Consequence: Overestimating area when using slant measurements.

Mistake 6: Ignoring Waste Factor

Consequence: Running short mid-project and facing dye lot mismatches.

Mistake 7: Not Accounting for Obstacles

• Subtract: Closets, islands, fireplaces, stairs
• Add: Bay windows, bump-outs, alcoves
• Measure each section separately

Consequence: Paying for materials you don't need or running short.

Mistake 8: Rounding Too Early

• Keep full precision during calculation
• Round only the final answer
• For materials: always round up

Consequence: Cumulative rounding errors on complex projects.

Verification Checklist:

1. Are all measurements in the same unit?
2. Did I use radius (not diameter) for circles?
3. Is the height perpendicular (not slant)?
4. Did I account for all sections (add-ons and cutouts)?
5. Have I added appropriate waste factor?
6. Does the result seem reasonable for the space?

Explore related calculators for complete project planning:

Geometry Calculators:

• Volume Calculator: Calculate 3D volume for concrete, mulch, soil, and fill materials. Area × depth = volume for rectangular shapes.

• Circle Calculator: Specialized calculations for circles including circumference, arc length, sector area, and chord calculations.

• Triangle Calculator: Solve triangles using any combination of sides and angles. Includes right triangle shortcuts and trigonometric functions.

Construction & Home Improvement:

• Flooring Calculator: Calculate flooring materials (hardwood, laminate, tile, carpet) based on room area. Includes waste factors, box counts, and cost estimates.

• Paint Calculator: Determine paint gallons needed based on wall area. Accounts for doors, windows, coats, and paint coverage rates.

• Tile Calculator: Calculate tile quantity, grout, and mortar based on floor or wall area. Includes pattern waste factors and box calculations.

Why Cross-Reference:

Project Planning Workflow:

1. Measure dimensions using this area calculator
2. Convert units if needed (sq ft to sq m, etc.)
3. Add waste factor based on project type
4. Calculate materials using specialized calculators
5. Estimate costs with local pricing

This integrated approach ensures accurate material ordering and budget planning for any project size.

Understanding land measurements for real estate and agriculture:

Acre Definition and Conversions:

Acre Visualization:

• Football field: ~1.32 acres (including end zones)
• Square acre: 208.7 × 208.7 feet
• Long narrow acre: 66 × 660 feet (chain × furlong)
• Baseball infield: ~0.02 acres

Common Lot Sizes:

Agricultural Land Measures:

Calculating Land Area from Dimensions:

Frontage vs. Depth:

Land value often depends on frontage (street/water access):

• Frontage: Width along road, waterfront, or access point
• Depth: Distance from frontage to rear boundary
• Flag lots: Narrow frontage with wider rear portion

A 1-acre lot could be:

• 43.5' × 1,000' (very narrow, limited use)
• 150' × 290' (typical rectangular lot)
• 209' × 209' (square, most versatile)

Survey Terminology: