About

Miscalculating area leads to material waste in construction, incorrect land valuations, and flawed engineering designs. This calculator implements exact closed-form formulas for 10 geometric shapes including Heron's formula for triangles from three sides: A = √s(s − a)(s − b)(s − c). Results are cross-converted across 10 area units from mm² to mi². Inputs persist across sessions. Note: all formulas assume Euclidean plane geometry and break down on curved surfaces or at astronomical scales.

Formulas

Each shape uses a distinct closed-form equation. All inputs are in the selected linear unit; output is in the corresponding squared unit.

Circle

A = π ⋅ r²

Rectangle

A = l × w

Triangle (base & height)

A = 12 ⋅ b ⋅ h

Triangle (Heron’s formula)

s = a + b + c2
A = √s(s − a)(s − b)(s − c)

Trapezoid

A = a + b2 ⋅ h

Ellipse

A = π ⋅ a ⋅ b

Parallelogram

A = b ⋅ h

Rhombus

A = d₁ ⋅ d₂2

Regular Polygon

A = n ⋅ s²4 ⋅ tan(π ÷ n)

Sector

A = θ360 ⋅ π ⋅ r²

Annulus (Ring)

A = π ⋅ (R² − r²)

Where: r = radius, l = length, w = width, b = base, h = height, a, b, c = side lengths, d₁, d₂ = diagonals, n = number of sides, s = side length, θ = central angle in degrees, R = outer radius, r = inner radius.

Reference Data

Unit	Symbol	Relative to 1 m²	Common Use
Square Millimeter	mm²	1,000,000	Precision engineering, electronics
Square Centimeter	cm²	10,000	Paper sizes, small objects
Square Meter	m²	1	Rooms, apartments, SI base
Square Kilometer	km²	0.000001	Cities, countries, large regions
Square Inch	in²	1,550.0031	US manufacturing, screens
Square Foot	ft²	10.76391	US real estate, flooring
Square Yard	yd²	1.19599	Carpet, landscaping (US/UK)
Acre	ac	0.000247105	Agriculture (US/UK)
Hectare	ha	0.0001	Agriculture (metric), land surveys
Square Mile	mi²	3.861×10⁻⁷	US geography, county areas
Are	a	0.01	European land plots
Barn	b	10²⁸	Nuclear cross-sections

Frequently Asked Questions

Heron's formula requires the triangle inequality to hold: the sum of any two sides must exceed the third side. If a + b ≤ c for any permutation, the three lengths cannot form a valid triangle and the expression under the radical becomes negative. Verify your measurements before computing.

The formula A = ns² ÷ (4 tan(π÷n)) is mathematically valid for any n ≥ 3, but geometrically only integer values produce a closed regular polygon. This calculator enforces integer values with a minimum of 3.

Conversions use IEEE 754 double-precision floating point with factors derived from NIST SP 811 standards. For example, 1 ft = 0.3048 m exactly (by definition). Results display up to 6 significant figures to avoid false precision.

No. Irregular polygons require vertex coordinates and the Shoelace formula (A = 12⋅|n∑i=1(x_iy_i+1 − x_i+1y_i)|). This tool handles regular polygons only, where all sides and angles are equal.

The calculator expects the central angle θ in degrees (0 - 360). If you have radians, multiply by 180π before entering. The equivalent radian formula is A = 12r²θ.

The annulus formula A = π(R² − r²) requires the outer radius R to be strictly greater than the inner radius r. If R ≤ r, the calculator will display a validation error.