Area Calculator

Area calculator for 10 common geometric shapes

This area calculator covers 10 shapes: circle, rectangle, square, triangle (base-height, Heron's formula, or two-sides-angle), trapezoid, parallelogram, ellipse, sector, rhombus, and regular polygon. Select a shape from the menu, enter the required dimensions, and the area (and perimeter where applicable) updates in real time. The formula used is shown below the result so you can verify or adapt it.

All calculation runs locally in JavaScript — your measurements are never sent to a server. No account, no upload, no signup required. Works on desktop and mobile, and stays functional offline once the page has loaded.

Essential area formulas — a quick reference

The most-used area formulas: Circle: A = π × r². Rectangle: A = l × w. Square: A = s². Triangle (base/height): A = ½ × b × h. Triangle (Heron's): A = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2. Trapezoid: A = ½ × (a+b) × h. Parallelogram: A = b × h. Ellipse: A = π × a × b. Sector: A = ½ × r² × θ (θ in radians). Rhombus: A = ½ × d₁ × d₂ (diagonals).

These formulas appear on every school syllabus from middle school through college geometry. Understanding which inputs are required — radius vs diameter, base vs side, height vs slant height — prevents the most common calculation errors. The labeled inputs in this tool remove that guesswork.

How to calculate square footage for flooring and rooms

The most common real-world use of rectangle area is calculating room square footage for flooring, carpeting, or painting. Measure the room in feet, multiply length by width, and that's your square footage. For an L-shaped room: split into two rectangles, calculate each, add them. For a room with a closet or alcove: add or subtract the sub-areas accordingly.

Flooring tip: always add 10% for waste from cuts and pattern matching. If your room is 180 sq ft, order 198 sq ft of flooring. For painting, divide wall area by the paint's coverage rate (typically 350–400 sq ft per gallon) to find how many gallons you need. Ceiling height × perimeter ÷ 400 = gallons of wall paint. This calculator handles the rectangle math instantly; the rest is arithmetic on the result.

Area vs. surface area — what's the difference?

Area refers to the flat, two-dimensional space inside a shape — a circle, triangle, or rectangle. Surface area is the total area of all faces of a three-dimensional object — a sphere, cube, or cylinder. For a sphere: SA = 4πr². For a cube with side s: SA = 6s². For a cylinder: SA = 2πr² + 2πrh (two circles plus the lateral surface).

This tool calculates flat 2D area only. For 3D surface area, dedicated calculators exist. The most practically important surface area calculations are painting a wall (flat area), wrapping a box (surface area of a rectangular prism = 2(lw + lh + wh)), and tiling a curved surface. Volume is a separate calculation again — a pool requires volume (length × width × depth), not area.

Comparing this to Wolfram Alpha, Google, and GeoGebra

Wolfram Alpha handles area queries via natural language and is accurate, but returns a single answer with minimal explanation of which inputs it used. Google's calculator handles simple cases (type 'area of circle radius 5' in search) but doesn't support shape switching or persistent labeled fields. GeoGebra is a full geometry suite — powerful for geometric construction but overkill for a quick area lookup.

This tool is built for speed: pick the shape, fill in the labeled fields, get area and perimeter instantly. The formula is displayed so you can check the work. No ads on the tool page, no login required, and no data collected. For classroom use, the formula display makes it appropriate for checking homework without just getting a black-box answer — students can see exactly which formula applied.

Where the formulas come from — derivation intuition

The area of a circle formula A = πr² is often treated as a fact to memorize, but it has an elegant informal proof. Slice a circle into a large number of thin pie sectors, then rearrange those sectors alternating point-up and point-down into a near-rectangle. The width of that rectangle approaches half the circumference (πr) and the height approaches the radius (r), giving area = πr × r = πr². The more sectors you cut, the closer the rearrangement gets to a perfect rectangle — this is the intuition behind integration in calculus.

The triangle formula A = ½ × base × height follows directly from the rectangle. Any triangle can be enclosed in a rectangle of the same base and height; the triangle always occupies exactly half of that rectangle's area. You can verify this by cutting any triangle in two along the altitude and flipping the pieces to form a rectangle. The factor of ½ is therefore not an arbitrary constant but a geometric necessity.

Heron's formula — A = √[s(s−a)(s−b)(s−c)] where s is the semi-perimeter — solves the harder problem of finding area when only the three side lengths are known. Its derivation combines the law of cosines (to express the altitude in terms of the sides) with algebraic factoring, ultimately collapsing into the compact radical form. In practice Heron's formula is indispensable for surveying and structural engineering where measuring a perpendicular height is impossible but all three sides are accessible.

Area units and why they scale by the square

A common source of error is forgetting that area units scale by the square of the length unit, not linearly. One metre equals 100 centimetres, but one square metre equals 10,000 square centimetres (100 × 100), not 100. Similarly, 1 km² = 1,000,000 m², and 1 ft² = 144 in². When converting, always square the linear conversion factor: to go from m² to cm², multiply by 100² = 10,000.

In land measurement, the acre is the dominant unit in the United States and United Kingdom. One acre equals exactly 43,560 square feet — a figure that traces back to the medieval definition of the amount of land one man with one ox could plough in a day (a furlong × a chain: 660 ft × 66 ft). A hectare is the metric equivalent used globally in agriculture and real-estate: 1 hectare = 10,000 m² ≈ 2.471 acres. Large US land tracts use the township-and-range survey system: a section is 1 square mile (640 acres), and a township is 36 sections (36 square miles).

For everyday construction, the key conversions are: 1 m² = 10.764 ft² and 1 ft² = 0.0929 m². Paint coverage is typically rated in ft² or m² per litre, so matching your room area to the right unit is essential before ordering. This calculator is unit-agnostic — results are in whatever squared unit your inputs used — so convert your raw measurements first if consistency matters.

Handling irregular shapes: decomposition, the shoelace formula, and GIS

No formula directly computes the area of an arbitrary irregular shape, but the decomposition method solves most practical cases. Break the irregular region into non-overlapping standard shapes — rectangles, right triangles, circles or semicircles — that together tile the whole area. Compute each sub-area individually and sum them. An L-shaped room, for example, splits into two rectangles; a room with a bay window adds a rectangle or semicircle to the main rectangle. The decomposition method works as long as the boundary can be approximated by straight lines and curves.

When you have a polygon defined by a list of vertex coordinates (x₁,y₁), (x₂,y₂), …, (xₙ,yₙ), the shoelace formula gives the exact area without breaking the polygon into triangles manually: A = ½ |Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|, where the last vertex wraps back to the first. The formula works for any simple (non-self-intersecting) polygon regardless of convexity. Surveyors and GIS analysts use it constantly — modern cadastral software computes parcel area this way from GPS-derived boundary coordinates.

For truly irregular curved boundaries — a pond, a field with meandering edges, a watershed — neither decomposition nor the shoelace formula is practical by hand. Mechanical planimeters historically traced the boundary and returned the enclosed area directly from gear rotations. Today, GIS tools (QGIS, ArcGIS, Google Earth Engine) compute area from digitised or GPS-traced polygons in any coordinate system, accounting for map projection distortion. For most construction and interior-design tasks, decomposition into rectangles and triangles is precise enough.

Composite shapes and real-world area problems

Composite shape problems combine addition and subtraction of areas. A circular pool (radius 3 m) set inside a rectangular deck (10 m × 8 m) leaves a surrounding area of 80 − π × 3² ≈ 80 − 28.27 = 51.73 m² for tiling. An annulus (ring) between two concentric circles with outer radius R and inner radius r has area π(R² − r²). Knowing which regions to add and which to subtract is the core skill; the individual formulas are just arithmetic once the decomposition is clear.

In flooring and tiling, the composite approach also handles waste planning. Calculate the net room area (gross floor minus fixed obstacles like pillars or hearths), then add the waste factor: 10% for straight-lay tile or wood, 15% for diagonal tile lay (more cuts at the edges), and up to 15–20% for natural stone with variable cuts. Multiply net area by 1.10 or 1.15 to get the quantity to order. For paint, measure each wall's area (height × width), sum all walls, then subtract doors (roughly 20 ft² each) and windows (roughly 15 ft² each). Divide by the paint's stated coverage — typically 350–400 ft² per gallon — and round up to the nearest whole gallon.

In real-estate and construction, gross floor area (GFA) includes all enclosed space measured to the exterior face of walls, while net floor area (NFA) excludes walls, columns, and non-usable spaces. The difference matters significantly in commercial leasing: a building with a GFA of 10,000 ft² may have an NFA of only 8,500 ft² once structural elements are excluded, and tenants pay rent on NFA. Residential listings in the US typically report the conditioned living area, which excludes garages, unfinished basements, and exterior walls — so two homes with identical external footprints can have very different listed square footages depending on finishing and measurement convention.