Area and Perimeter Calculator
Calculate the area and perimeter of common geometric shapes including rectangles, triangles, and polygons
Results
Area
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Perimeter
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Formula Used
Additional Properties
Area & Perimeter Formulas
Area measures the two-dimensional space enclosed by a shape, while perimeter measures the total length of the boundary. These are fundamental concepts in geometry used in fields from architecture and engineering to land surveying and everyday problem solving.
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | A = length × width | P = 2(length + width) |
| Square | A = side² | P = 4 × side |
| Triangle | A = ½ × base × height | P = a + b + c |
| Circle | A = πr² | C = 2πr |
| Ellipse | A = π × a × b | P ≈ π[3(a+b) - √((3a+b)(a+3b))] |
| Trapezoid | A = ½(base1 + base2) × height | P = a + b1 + b2 + d |
| Parallelogram | A = base × height | P = 2(a + b) |
| Rhombus | A = ½ × d1 × d2 | P = 4 × side |
| Regular Polygon | A = ¼ × n × s² × cot(π/n) | P = n × s |
Practical Applications
Construction & Architecture
- • Calculating flooring, paint, or tiling area for rooms
- • Determining fencing length for property boundaries
- • Estimating roof surface area for shingles
- • Planning garden bed layouts and walkways
Land Surveying
- • Measuring property lot areas for real estate
- • Calculating field areas for agriculture
- • Mapping park and recreation zones
- • Estimating land cost based on acreage
Science & Engineering
- • Cross-sectional area in fluid dynamics
- • Surface area calculations in heat transfer
- • Material requirements in manufacturing
- • Structural load distribution analysis
Key Concepts
Area
Area is the quantity that expresses the extent of a two-dimensional region on the plane. It is measured in square units (e.g., m², ft²). For simple shapes, area can be computed using well-known formulas. For irregular shapes, techniques like integration or grid counting are used.
Perimeter
Perimeter is the total distance around the boundary of a two-dimensional shape. For polygons, it is the sum of all side lengths. For curved shapes like circles, the perimeter is called the circumference. Perimeter is measured in linear units (e.g., m, ft).
Isoperimetric Inequality
Among all shapes with a given perimeter, the circle encloses the greatest area. This principle, known as the isoperimetric inequality, is why bubbles form spheres and why circular designs are efficient for enclosing maximum area with minimum boundary material.
References
The formulas and methods used in this calculator are based on established mathematical principles from the following sources:
Related Calculators
This calculator provides mathematical calculations for area and perimeter of standard geometric shapes based on well-established formulas. Results are accurate to four decimal places. The ellipse perimeter uses Ramanujan's approximation. While we strive for accuracy, please verify critical calculations independently. This tool is for educational and informational purposes.
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