Area Calculator
Calculate area of common 2D shapes — rectangle, triangle, circle, and more.
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Area Formulas for Common Shapes
Area measures the amount of 2D space enclosed by a shape. It is expressed in square units (cm², m², ft², in², etc.). Here are the key formulas:
Rectangle and Square
Rectangle: A = length × width. Square: A = side². For a 5×8 rectangle: A = 40 square units.
Triangle
A = ½ × base × height (perpendicular height). Also: A = ½ × a × b × sin(C) for two sides and included angle. Heron's formula for three sides: A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2.
Circle
A = π × r² (radius squared). Circumference = 2πr. For r=5: A = π×25 ≈ 78.54 square units.
Trapezoid
A = ½ × (base₁ + base₂) × height. The average of the two parallel bases times the perpendicular height.
Parallelogram and Ellipse
Parallelogram: A = base × height (perpendicular height). Ellipse: A = π × a × b (semi-major × semi-minor axes).
Area calculations are fundamental in construction (floor space), agriculture (field size), architecture (roof area), and manufacturing (material required). Converting between units: 1 ft² = 144 in², 1 m² = 10,000 cm².
Worked Example: Trapezoid Room Floor
A room has a trapezoidal floor with parallel sides of 9 m and 6 m, and a perpendicular height of 4 m. Calculate the floor area:
A = ½ × (base₁ + base₂) × height = ½ × (9 + 6) × 4 = ½ × 15 × 4 = 30 m². If flooring costs $25 per m², the total cost is 30 × $25 = $750.
Area Formulas Quick Reference
| Shape | Formula | Example |
|---|---|---|
| Rectangle | l × w | 10 × 6 = 60 |
| Square | s² | 5² = 25 |
| Triangle | ½ × b × h | ½ × 8 × 5 = 20 |
| Circle | π × r² | π × 7² ≈ 153.94 |
| Trapezoid | ½(b₁+b₂) × h | ½(10+6) × 4 = 32 |
| Parallelogram | b × h | 8 × 5 = 40 |
| Ellipse | π × a × b | π × 7 × 4 ≈ 87.96 |
Frequently Asked Questions
A = length × width. For a 6×4 rectangle: A = 24 square units.
A = π × 7² = π × 49 ≈ 153.94 square units.
A = ½ × base × height. The height must be perpendicular to the base. For base=8, height=5: A = 20 square units.
For a triangle with sides a, b, c: s=(a+b+c)/2, A=√(s(s-a)(s-b)(s-c)). Useful when height is unknown.
Square units: cm², m², km², in², ft², yd², acres (43,560 ft²), hectares (10,000 m²).
Divide the irregular shape into known shapes (rectangles, triangles, circles), calculate each area separately, then add them together. Alternatively, use the coordinate geometry shoelace formula if you know the vertices.
Area measures the space inside a flat 2D shape. Surface area measures the total area of all outer faces of a 3D solid. A cube with side 4 has face area 4² = 16 per face, and total surface area = 6 × 16 = 96 square units.