Area Calculator Calculators

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Area is the measure of the two-dimensional space enclosed within a shape, expressed in square units (cm², m², ft², acres, hectares). Calculating area is one of the most fundamental operations in mathematics, applied in construction, land survey, biology (leaf area, cell cross-section), ecology (habitat patches), and everyday life (flooring, painting). Different shapes have different area formulas derived from geometric principles. The most common shapes encountered in science and daily applications include rectangles, triangles, circles, trapezoids, and ellipses.

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Area Formulas for Common Shapes

  • Rectangle / Square: A = l × w (length × width)
  • Triangle: A = ½ × base × height
  • Circle: A = π × r² (r = radius)
  • Ellipse: A = π × a × b (a, b = semi-major and semi-minor axes)
  • Trapezoid: A = ½ × (a + b) × h (a, b = parallel sides; h = height)
  • Parallelogram: A = base × height
  • Regular hexagon: A = (3√3/2) × s² (s = side length)

Area of a Triangle (Heron's Formula)

When only side lengths are known: A = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 (semi-perimeter) and a, b, c are side lengths. Useful when the height is not directly measurable.

Unit Conversions

  • 1 m² = 10,000 cm²
  • 1 hectare = 10,000 m²
  • 1 acre = 4,047 m² ≈ 0.4047 ha
  • 1 ft² = 0.0929 m²
  • 1 mile² = 640 acres = 2.59 km²

Irregular Areas

For irregular shapes: divide into regular sub-shapes and sum their areas; use the shoelace formula for polygon with known vertex coordinates: A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|; or use digital image analysis (ImageJ) for biological specimens like leaves.

Glossary

Area
The two-dimensional space enclosed within a shape, measured in square units (m², cm², acres, hectares); calculated using shape-specific formulas.
Heron's Formula
A = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2; calculates triangle area from the three side lengths when height is unknown.
Hectare (ha)
A unit of area equal to 10,000 m² (100 m × 100 m); 1 hectare = 2.471 acres; the standard unit for agricultural land area and forest inventory.

Frequently Asked Questions

Area of a circle = π × r², where r is the radius (half the diameter). Example: circle with radius 5 cm: A = π × 5² = π × 25 = 78.54 cm². If you know the diameter (d) instead: r = d/2, so A = π × (d/2)². Example: d = 10 cm: A = π × 25 = 78.54 cm². Use π ≈ 3.14159 or the π key on your calculator for accurate results.

If you know base and height: A = ½ × base × height. Example: base = 8 m, height = 5 m: A = ½ × 8 × 5 = 20 m². If you know all three sides (a, b, c) but not the height, use Heron's formula: s = (a+b+c)/2; A = √[s(s−a)(s−b)(s−c)]. Example: sides 3, 4, 5 m: s = 6; A = √[6×3×2×1] = √36 = 6 m².

Key conversions: 1 m² = 10,000 cm² = 10.764 ft². 1 hectare = 10,000 m² = 2.471 acres. 1 acre = 4,047 m² = 43,560 ft². 1 km² = 100 hectares = 247.1 acres. To convert m² to cm²: multiply by 10,000. To convert cm² to m²: divide by 10,000. For land measurements: 1 football field ≈ 0.53 acres ≈ 0.21 hectares.

Three main approaches: (1) Decompose — divide the shape into recognizable sub-shapes (rectangles, triangles, circles), calculate each area, and sum them. (2) Shoelace formula — for polygons with known vertex coordinates: A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|. (3) Digital analysis — scan or photograph the shape, calibrate with a scale bar, and use software like ImageJ to measure pixel area (widely used for leaf area index and cell cross-sections in biology).