Area Calculator,
six shapes, instant results.

Geometry guide

Area formulas: every shape explained.

Area is the amount of two-dimensional space enclosed within a boundary. It is measured in square units: square metres (m²), square centimetres (cm²), square feet (ft²), and so on. Every polygon and curved shape has a deterministic formula that converts its linear dimensions into an area, and this calculator implements all six of the most common ones.

Square: A = s²

A square has four equal sides and four right angles. Its area is the side length multiplied by itself — a direct application of the definition of a square number.

where s is the side length. Perimeter: P = 4s.

A square with side 5 m has area 25 m² and perimeter 20 m. Because all sides are equal, a square maximises the enclosed area for a given perimeter among all rectangles. This is why square-ish rooms feel more spacious than long, thin ones of the same area.

Rectangle: A = l × w

A rectangle has four right angles and two pairs of equal parallel sides. Its area is simply length times width — the fundamental definition of area that every other formula is derived from or compared to.

Perimeter: P = 2(l + w).

Rectangles are the most common shape in architecture and engineering. Floor area, screen area, page area, all are rectangular products.

Circle: A = π · r²

A circle encloses the maximum possible area for a given perimeter (among all plane figures) — the isoperimetric inequality. The area formula is derived by integrating thin concentric rings from radius 0 to r:

Circumference: C = 2π · r ≈ 6.2832 · r.

π ≈ 3.14159. A circle with radius 5 m has area 78.54 m², compared to a square with the same perimeter (side ≈ 7.85 m, area ≈ 61.7 m²), the circle encloses 27% more area.

Triangle: A = ½ · b · h

The area of any triangle is half the product of its base and perpendicular height, regardless of orientation or type (acute, right, or obtuse). This follows directly from the rectangle formula: any triangle is exactly half of a rectangle with the same base and height.

where b is the base length and h is the perpendicular height (altitude) from the base to the opposite vertex.

Perimeter requires all three side lengths. For an isosceles triangle (two equal sides), the non-base sides each equal √(h² + (b/2)²), so P = b + 2·√(h² + (b/2)²). This calculator uses the isosceles assumption when computing perimeter from base and height.

If you know all three sides, use Heron’s formula instead:

Trapezoid: A = ½ · (b₁ + b₂) · h

A trapezoid (called a trapezium in British English) has exactly one pair of parallel sides, called the bases. Its area is the average of the two bases multiplied by the perpendicular height between them. This formula is a generalisation: when b₁ = b₂ it reduces to the rectangle formula; when b₂ = 0 it reduces to the triangle formula.

Perimeter with known leg length c: P = b₁ + b₂ + 2c. For an isosceles trapezoid (equal legs), each leg = √(h² + ((b₁ − b₂)/2)²).

Ellipse: A = π · a · b

An ellipse is a stretched circle, parameterised by its semi-major axis a (half the longest diameter) and semi-minor axis b (half the shortest diameter). Its area formula is an elegant generalisation of the circle:

When a = b = r this reduces to πr² (the circle). Eccentricity e = √(1 − (b/a)²) measures how “stretched” the ellipse is: a circle has e = 0; a very elongated ellipse approaches e = 1.

Unlike the circle, there is no exact closed-form formula for the perimeter of an ellipse. This calculator uses Ramanujan’s first approximation:

This approximation is accurate to within 0.02% for all ellipses, which is sufficient for engineering and everyday use.

Unit conversion: area scales with the square

A common mistake is applying linear unit factors to area calculations. 1 metre = 100 centimetres, but 1 m² ≠ 100 cm². The correct relationship is:

Similarly, 1 ft² = 144 in², and 1 m² ≈ 10.764 ft². This calculator handles all conversions automatically, so you can enter in feet and read the result in cm² without doing any unit arithmetic.

Comparison of areas for equal dimensions

For a given linear dimension s, the six shapes rank by area as follows (largest to smallest):

Real-world applications

• Flooring and tiling: Measure the room as a rectangle (or sum of rectangles), compute the area in m² or ft², and add 10% waste allowance for cuts.
• Painting: Wall area is a rectangle (width × height). Subtract window and door areas (rectangles). One litre of paint covers 10–12 m².
• Landscaping: Circular garden beds use πr²; trapezoidal plots use the trapezoid formula.
• Engineering cross-sections: Pipe cross-sections are circles (πr²); I-beam flanges are rectangles.
• Elliptical pools and fields: Ellipses appear in stadium tracks, architectural domes, and irrigated crop circles.

Worked examples

Circular swimming pool: radius = 3.5 m. A = π × 3.5² ≈ 38.48 m². Surface area for a pool cover: 38.48 m². Circumference for tiling: ~21.99 m.

Trapezoidal garden plot: b₁ = 12 m, b₂ = 8 m, h = 5 m. A = ½ × (12 + 8) × 5 = 50 m².

Triangular roof section: base = 9 m, height = 3.5 m. A = ½ × 9 × 3.5 = 15.75 m². Multiply by two for a gable roof with equal slopes.