Find the area and perimeter of any common 2D shape — square, rectangle, circle, triangle, trapezoid, parallelogram, ellipse, rhombus, or circle sector.
Pick a shape and enter its dimensions — area updates live.
Area
Perimeter
20 cm
distance around
Area (m²)
0.0025
SI cubic-area unit
Area (ft²)
0.0269
square feet
Area (in²)
3.875
square inches
| Shape | Area formula | Perimeter formula |
|---|---|---|
| Square | A = a² | P = 4a |
| Rectangle | A = l · w | P = 2(l + w) |
| Circle | A = πr² | C = 2πr |
| Triangle | A = ½ · b · h | P = a + b + c |
| Trapezoid | A = ½(a + b)h | P = sum of sides |
| Parallelogram | A = b · h | P = 2(a + b) |
| Ellipse | A = πab | P ≈ π(3(a+b) − √((3a+b)(a+3b))) |
| Rhombus | A = ½ · d₁ · d₂ | P = 4a |
| Sector | A = ½ · r² · θ | P = 2r + rθ |
The Method
For any 2D shape, area is the integral of 1 over the enclosed region. For familiar shapes the integral collapses to a clean closed-form expression: πr² for a circle, l·w for a rectangle, and ½·b·h for a triangle. The trapezoid and parallelogram both extend the rectangle's base × height idea, and Heron's formula handles a triangle when you only know the three side lengths. This calculator evaluates closed-form formulas directly so the result is exact within floating-point precision.
Working for selected shape
About This Tool
An area calculator finds the amount of 2D space enclosed by a shape. Pick one of nine common shapes — square, rectangle, circle, triangle, trapezoid, parallelogram, ellipse, rhombus, or circle sector — enter its dimensions, and the calculator returns the exact area in the square unit you choose, along with the perimeter (or circumference) and quick conversions to m², ft², and in².
Area is foundational in every domain that touches geometry. Construction uses it for flooring, paint and tiling estimates; real estate uses it for floor plans; agriculture uses it to size fields; and physics uses it inside surface integrals and the definition of pressure. Every standard formula here comes from the same idea — integrating one over the enclosed region — but for the nine common shapes the answer simplifies to a closed-form expression you can evaluate without calculus.
The calculator uses textbook formulas: A = a² for a square, A = l·w for a rectangle, A = πr² for a circle, A = ½·b·h for a triangle, A = ½(a+b)h for a trapezoid, A = b·h for a parallelogram, A = πab for an ellipse, A = ½·d₁·d₂ for a rhombus, and A = ½·r²·θ for a circle sector. Every formula is evaluated with double-precision arithmetic accurate to about 15 significant digits.
Use this free area calculator for homework, DIY projects, gardening, science class, or any time you need a quick 2D area. All calculation is local — no sign-up, no tracking, no data sent to any server.
Nine Common Shapes
Square, rectangle, circle, triangle, trapezoid, parallelogram, ellipse, rhombus, sector.
Six Length Units
Metres, centimetres, millimetres, feet, inches, yards — converted on the fly.
Area + Perimeter
Both fundamental quantities reported side by side.
Live Diagram
Scaled SVG of the current shape with labelled dimensions.
100% Free & Private
No account, no tracking — every calculation runs locally in your browser.
Step-by-Step Working
Each formula substitution shown — perfect for homework or revision.
From shape pick to perimeter in under a minute.
Choose from the chips at the top — the calculator rebuilds the required input fields for that shape.
Pick from metres, centimetres, millimetres, feet, inches, or yards. The result is given in the corresponding square unit.
Fill in each required length, radius, or angle. Inputs accept decimals and update the area in real time.
The headline shows your area in the chosen unit. Stat tiles convert it to m², ft², and in².
The shape diagram is scaled to your inputs so you can visually verify the dimensions you entered.
The formula card shows the full substitution — useful for homework write-ups and verifying a hand calculation.
Everything you need to know about 2D area and perimeter formulas.
Area is the amount of two-dimensional space enclosed by a closed curve, measured in square units — m², cm², ft², and so on. It answers the question: how much surface does this shape cover?
A circle of radius r has area A = πr². For example a circle with radius 5 has area π·25 ≈ 78.54 square units. This is one of the oldest known formulas, attributed to Archimedes.
Area is the space inside a shape (2D, square units); perimeter is the distance around the outside (1D, length units). Two shapes with the same area can have very different perimeters — a long thin rectangle has more perimeter than a square of equal area.
If you know the base and height, A = ½ · base · height. If you know all three sides, use Heron's formula: A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2 is the semi-perimeter. Both give the same answer.
An ellipse with semi-major axis a and semi-minor axis b has area A = πab. When a = b = r it reduces to a circle. Perimeter has no closed-form expression; this calculator uses Ramanujan's approximation, which is accurate to about 0.04% for typical ellipses.
Whatever fits your problem — cm² or in² for everyday objects, m² or ft² for rooms and land, mm² for small components. The calculator handles all six standard units and shows quick equivalents.
Area scales with the square of linear size. Double a shape's lengths and area becomes 4×; triple them and area becomes 9×. This is why a 30 cm pizza isn't twice the size of a 15 cm pizza — it's four times.
A sector is a pie slice — the region bounded by two radii and an arc. For radius r and central angle θ (in radians): A = ½·r²·θ. With θ in degrees, A = (θ/360) × πr².
All formulas are evaluated in IEEE 754 double-precision floating-point — accurate to about 15-17 significant digits, far beyond any real-world measurement.