Area of Shapes Calculator
Calculate the area of any 2D shape — circle, rectangle, triangle, trapezoid, parallelogram, ellipse, and more.
Supports metric and imperial units.
Area Formulas for Common 2D Shapes
Area is the amount of two-dimensional space enclosed within a flat shape. All area results are in square units (e.g., cm², m², in², ft²).
Rectangle
A = length × width
The most fundamental area formula. A square is a special rectangle where length = width.
Square
A = side²
Circle
A = π × r²
The area of a circle grows with the square of the radius — doubling the radius quadruples the area. The connection between π and the circle area was known to the ancient Egyptians and Babylonians, though Archimedes provided the first rigorous mathematical proof around 250 BC.
Triangle
A = (1/2) × base × height
The height must be perpendicular to the base. For any triangle (including obtuse), you can always drop a perpendicular from the opposite vertex.
Trapezoid (Trapezium)
A = (1/2) × (a + b) × height
Where a and b are the lengths of the two parallel sides. Think of it as averaging the two parallel sides, then multiplying by height.
Parallelogram
A = base × height
The height is the perpendicular distance between the parallel sides — not the slant side length.
Ellipse
A = π × a × b
Where a and b are the semi-major and semi-minor axes. A circle is a special case of an ellipse where a = b = r.
Regular Hexagon
A = (3√3 / 2) × side²
A regular hexagon can be divided into 6 equilateral triangles, each with area (√3/4) × s².
Regular Pentagon
A = (√(25 + 10√5) / 4) × side²
Approximately A ≈ 1.720 × s².
Rhombus
A = (d₁ × d₂) / 2
Where d₁ and d₂ are the lengths of the two diagonals.
Mnemonic Tips
- Rectangle: “Length times width” — think of counting tiles
- Triangle: “Half of a rectangle” — a triangle is exactly half a rectangle with the same base and height
- Circle: “Apple pie are squared” — a popular mnemonic for A = πr²
- Trapezoid: “Average the parallel sides, multiply by height”
Unit Conversion Reference
| Unit | Square Conversion |
|---|---|
| 1 ft² | 144 in² |
| 1 m² | 10,000 cm² |
| 1 acre | 43,560 ft² |
| 1 hectare | 10,000 m² |
Real-World Applications
- Flooring: Calculate square footage needed for hardwood or tile
- Paint: One gallon covers approximately 350 ft² — how many gallons to paint a room?
- Landscaping: Sod, mulch, and seed are sold by area
- Real estate: Land plots and building footprints measured in square meters or acres
- Fabric: Pattern cutting in dressmaking requires accurate area calculations
- Solar panels: Panel output and installation cost scale directly with roof area