Area Calculator
Calculate the area of a shape, room, irregular polygon, property or 3D solid. Choose what you are measuring, enter the dimensions in any supported units, and see the formula, substitution, conversions and verification.
One calculator for every legitimate meaning of area — without mixing incompatible formulas or hiding how the result was obtained.
Result
Verified Splitting the rectangle along its diagonal into two triangles and summing their Heron areas reproduces the product (20 vs 20).
- Perimeter
- 18 m
- Diagonal
- 6.4031242 m
- Aspect ratio
- 1.25 : 1
- Original inputs
- length = 4 m · width = 5 m
- Normalized
- length = 4 m (4 m) · width = 5 m (5 m)
- Method
- Rectangle — Length and width
- Formula
A = l × w- Substitution
- A = 4 × 5 = 20
- Steps
- Multiply the two perpendicular sides: 4 × 5 = 20.
- Decimal result
- 20 m²
- Assumptions
- Dimensions are normalized to meters before the formula is applied; only the resulting area is converted to m².
- Rounding
- Auto rounding (up to 6 significant decimals shown); internal values are unrounded.
- Engine
- area-engine v1.0.0 · state v1 · receipt 1.0.0
- Share id
- a8c2a655b6e6cf8d
Area formulas at a glance
| Shape | You need | Formula | Try it |
|---|---|---|---|
| Rectangle | length l, width w | A = l × w |
|
| Square | side s (or diagonal, perimeter, area) | A = s² = d²/2 |
|
| Triangle | base b and perpendicular height h | A = b·h/2 |
|
| Triangle (3 sides) | sides a, b, c | p=(a+b+c)/2 · A=√(p(p−a)(p−b)(p−c)) |
|
| Circle | radius r (or d, C, A) | A = πr² |
|
| Circular sector | radius r, angle θ | A = (θ/360°)·πr² |
|
| Semicircle | radius r | A = πr²/2 |
|
| Annulus (ring) | outer R, inner r | A = π(R² − r²) |
|
| Ellipse | semi-axes a, b | A = πab |
|
| Parallelogram | base b, perpendicular height h | A = b·h = a·b·sin θ |
|
| Trapezoid | parallel sides a, b and height h | A = ((a+b)/2)·h |
|
| Rhombus | diagonals d₁, d₂ | A = d₁·d₂/2 = b·h = s²·sin θ |
|
| Kite | diagonals d₁, d₂ | A = d₁·d₂/2 |
|
| Regular polygon | n sides of length s | A = n·s²/(4·tan(π/n)) = P·a/2 |
How to calculate area
Identify the shape, measure the dimensions its formula needs, and multiply them in consistent units. A rectangle needs its two perpendicular sides (A = l × w); a triangle needs a base and the height measured at a right angle to it (A = ½bh); a circle needs only its radius (A = πr²). The one universal rule is that area multiplies two lengths, so the answer is always in square units — and mixed input units must be reconciled before the formula runs, which this calculator does by normalizing every dimension to meters first.
How to calculate square footage
Measure the room's length and width in feet (a tape reading like 12 ft 6 in is 12.5 ft, since 6 inches is half a foot), then multiply: 12.5 × 9 = 112.5 ft². For an L-shaped or interrupted floor, split it into rectangles, add them, and subtract openings like stairwells with the Multiple areas workflow. One square foot is exactly 0.09290304 m², so 112.5 ft² is 10.451592 m².
How to calculate an irregular area
There are two honest strategies. Decomposition: cut the outline into rectangles, triangles and circle parts, add the pieces, and subtract any holes — the composite builder tracks each contribution. Coordinates: walk the boundary, record each corner as an (x, y) pair, and apply the Shoelace formula, which sums the cross-products of consecutive vertices. Both show their work in the calculation receipt.
Area from coordinates
The Shoelace formula, A = ½ × |Σ(xᵢ·yᵢ₊₁ − yᵢ·xᵢ₊₁)|, computes the exact area of any simple (non-self-intersecting) polygon from its vertex list, in either winding direction. This calculator validates the outline first — at least three distinct points, finite coordinates, no crossing edges — then reports area, perimeter, centroid, bounding box and orientation, and cross-checks the result with an independent ear-clipping triangulation.
Measuring land on a map
Latitude/longitude vertices cannot go straight into the planar Shoelace formula — degrees of longitude shrink toward the poles. This calculator uses the Chamberlain–Duquette spherical-excess method on the WGS84 authalic sphere (declared in every receipt), with boundary lengths from haversine great-circle distances. For typical parcels the earth-model error is a small fraction of a percent; the dominant uncertainty is where you place the points, which is why map results are estimates and never a substitute for a survey.
Area vs perimeter vs surface area vs volume
These four measure different things. Perimeter is the length around a flat figure (meters). Area is the flat surface it encloses (square meters). Surface area is the total area of a solid's faces and curved skin (square meters — a cylinder's two ends plus its wrapped side). Volume is the space inside a solid (cubic meters — see the Volume Calculator). This page computes areas and surface areas; the Circle Calculator and Triangle Calculator solve those figures completely, including angles and missing sides.
Methodology & verification
The workspace runs five modes on one dedicated engine stack (area-engine v1.0.0). Every dimension is parsed (including feet-and-inches input), normalized to meters BEFORE the geometric formula runs, and only the resulting area is converted to the output unit, so mixed units like 12 ft × 3 m can never double-apply a conversion factor. Plane shapes use their standard formulas (l×w, s², ½bh, Heron, ½ab·sin C, πr², sector fraction, π(R²−r²), πab, bh, ½(a+b)h, ½d₁d₂, ns²/(4tan(π/n))); coordinate outlines use the Shoelace formula cross-checked by ear-clipping triangulation; land polygons use the Chamberlain–Duquette spherical-excess method on the WGS84 authalic sphere with haversine boundary lengths; 3D solids separate lateral, base and total surfaces. Each result ships a calculation receipt with the formula, substitution, steps, exact π form where applicable, conversions and an explicitly labeled independent verification — methods with no honest second path say so instead of pretending. Map results are estimates, not surveys.
- Unit basis: exact international factors (1 in = 0.0254 m, 1 ft = 0.3048 m, 1 yd = 0.9144 m, 1 mi = 1609.344 m, 1 acre = 4046.8564224 m²); dimensions normalize to meters before any formula, and only the finished area is converted.
- Precision policy: display rounding (auto, 0–10 decimal places, or 3–12 significant figures) never touches internal values, conversions or verification; the receipt records the rule applied.
- Polygon algorithm: Shoelace formula with normalization (duplicate and closing vertices removed), self-intersection rejection, and an independent ear-clipping triangulation cross-check.
- Geographic method: Spherical-excess line integral (Chamberlain–Duquette, JPL 07-3) on the WGS84 authalic sphere (R = 6371007.1809 m); perimeters use haversine great-circle edges; antimeridian crossings and holes are handled explicitly.
- Verification labels: “Verified” appears only when a genuinely different computation path reproduces the result (decomposition, coordinate reconstruction, independent formula, quadrature, or a local-projection cross-check). Methods with no honest second path say “independent check unavailable”.
- Map limitations: land results are estimates on a spherical earth model, dependent on your point placement — never survey-grade and never a legal boundary.
Author: Ugo Candido · Last reviewed: · Engine: area-engine v1.0.0 · Every calculation runs in your browser; dimensions are never sent to a server. Map tiles and optional place search contact the configured map provider only after you open the map. See the public changelog.
Worked examples
Each button loads the example into the workspace above with its full receipt.
Frequently asked questions
How do I calculate area?
Pick the mode that matches what you are measuring, choose the shape or method, and enter the dimensions. The calculator applies the standard formula for that figure, shows the substitution with your numbers, and reports the area in your chosen square unit.
Is area always length times width?
No — that is the rectangle formula only. Triangles use half of base × height, circles use πr², trapezoids average their parallel sides, and irregular outlines need the Shoelace formula or decomposition into simpler shapes. The calculator selects the right formula for the figure you choose.
How do I calculate square footage?
Measure the length and width, enter them in the Square Footage mode (feet-and-inches input like 12 ft 6 in works directly), and read the result in square feet. For example, 12 ft 6 in × 9 ft = 112.5 ft², with square meters, square yards and acres shown alongside.
Can length and width use different units?
Yes. Each dimension has its own unit selector, so 12 ft × 3 m is valid. Both values are converted to meters before multiplying, and only the finished area is converted to the output unit — a conversion factor is never applied twice.
How do I calculate an irregular area?
Either build the outline from simple components (rectangles, triangles, circles) and subtract the cutouts, or enter the corner coordinates and let the Shoelace formula measure the enclosed polygon. Both approaches show every component or vertex in the receipt.
How accurate is map area measurement?
The math uses a declared spherical-excess method on the WGS84 authalic sphere, which is within a fraction of a percent of a rigorous ellipsoidal result for typical parcels — but the dominant error is where you place the points. Map results are estimates, not legal boundaries, and never replace a survey.
What is the difference between area and perimeter?
Area measures the surface enclosed by a figure, in square units; perimeter measures the length of the boundary around it, in plain length units. Two shapes can share a perimeter and enclose very different areas.
What is the difference between area and surface area?
Area applies to a flat, two-dimensional figure. Surface area is the total area of every face or curved skin of a three-dimensional solid — a cylinder's two circular ends plus its curved side, for example. Volume, the space inside, is a third quantity with cubic units.
How do I subtract doors, openings or cutouts?
Add the opening as its own row and mark it Subtract. The calculator totals the added rows, subtracts the cutout rows, and shows each contribution separately — a stair opening of 6 ft × 3 ft removes exactly 18 ft² from the total.
Why is area measured in square units?
Because area multiplies two lengths. Meters times meters gives square meters; feet times feet gives square feet. That is also why converting an area needs the squared conversion factor: 1 ft = 0.3048 m, but 1 ft² = 0.09290304 m².
How many decimal places should I use?
For rooms and materials, one or two decimals is usually plenty — tape-measure input is not more precise than that. The precision control offers automatic rounding, 0–10 decimal places or 3–12 significant figures, and the receipt records which rule was applied; internal math always stays unrounded.
Can I calculate a missing length from a known area?
Yes. The rectangle shape offers solve-backward methods: enter the known area plus one side to get the other side (w = A ÷ l), or a diagonal plus one side. The square shape likewise recovers the side from a known area, diagonal or perimeter.