Free Area Calculator
Calculate the area of any geometric shape instantly. This free online area calculator supports circles, rectangles, triangles, trapezoids, parallelograms, ellipses, and sectors with step-by-step solutions and unit conversions.
What is Area?
Area is the total amount of two-dimensional space enclosed within the boundary of a shape. It measures the surface covered by a flat figure and is expressed in square units such as square meters (m²), square feet (ft²), or square centimeters (cm²).
Area is one of the most fundamental concepts in geometry. Whether you are measuring a room for flooring, calculating land for a property, or determining the surface area of a wall for painting, area calculations are part of everyday problem-solving.
Unlike perimeter, which measures the distance around a shape, area measures the space inside it. A rectangle with a length of 10 m and a width of 5 m has a perimeter of 30 m but an area of 50 m².
Area Formulas for Every Shape
Each shape has its own formula based on its geometric properties. Here is a complete reference:
| Shape | Formula | What you need to measure |
|---|---|---|
| Circle | A = π × r² | Radius |
| Rectangle | A = l × w | Length and width |
| Triangle | A = ½ × b × h | Base and height |
| Trapezoid | A = ½ × (a + b) × h | Two parallel sides and height |
| Parallelogram | A = b × h | Base and perpendicular height |
| Ellipse | A = π × a × b | Two semi-axes |
| Sector | A = ½ × r² × θ | Radius and angle in radians |
For the sector formula, θ must be in radians. To convert degrees to radians, use the formula: θ (radians) = θ (degrees) × π / 180. For example, 90° = π/2 ≈ 1.5708 radians.
How to Use This Area Calculator
- Select a shape from the grid at the top of the calculator.
- Enter the required dimensions in the input fields. The fields change automatically based on the selected shape.
- Choose your preferred unit from the unit selector (mm², cm², m², km², in², ft², yd², acres, hectares).
- View the result card showing the area value, the formula used, and step-by-step calculation.
- Check the unit conversions section to see the area in different units.
The visual diagram updates in real time to show the labeled dimensions of the selected shape, making it easy to verify your inputs.
Features
- 7 shape types supported: circle, rectangle, triangle, trapezoid, parallelogram, ellipse, and sector
- Step-by-step calculation showing every intermediate value
- Visual shape diagrams with labeled dimensions that update in real time
- 9 area units with automatic conversion: mm², cm², m², km², in², ft², yd², acres, hectares
- Formula display for each shape so you learn while you calculate
- Instant results as you type, no need to click a calculate button
- Mobile-friendly responsive design that works on any screen size
Area Units Conversion Table
Different industries and regions use different area units. This table shows the relationships between common units:
| Unit | Equivalent in m² | Common Use |
|---|---|---|
| 1 mm² | 0.000001 m² | Engineering, electronics |
| 1 cm² | 0.0001 m² | Science, small objects |
| 1 m² | 1 m² | Construction, flooring |
| 1 km² | 1,000,000 m² | Geography, land surveys |
| 1 in² | 0.00064516 m² | US manufacturing |
| 1 ft² | 0.092903 m² | US real estate, interiors |
| 1 yd² | 0.836127 m² | Landscaping, fabric |
| 1 acre | 4,046.86 m² | Agriculture, property |
| 1 hectare | 10,000 m² | International agriculture |
To convert from one unit to another, multiply by the ratio of the conversion factors. For example, to convert 100 ft² to m²: 100 × 0.092903 = 9.29 m².
Real-World Applications
Land Measurement
Surveyors and real estate professionals use area calculations to determine property boundaries, lot sizes, and land values. A rectangular plot measuring 50 m × 30 m has an area of 1,500 m² (approximately 0.37 acres).
Flooring and Tiling
When installing flooring, you need the room's area to calculate material requirements. For an L-shaped room, you can split it into two rectangles, calculate each area separately, and add them together. Always add 10% extra for waste and cuts.
Painting and Wall Coverage
Paint manufacturers specify coverage per unit area (typically 1 gallon covers about 350-400 ft²). Calculate the wall area by multiplying height × width for each wall, then subtract window and door areas.
Farming and Agriculture
Farmers measure fields in acres or hectares to plan crop yields, fertilizer application, and irrigation. A hectare of wheat might yield 3-4 tonnes depending on conditions.
Construction and Architecture
Architects calculate floor areas for building permits, energy efficiency ratings, and cost estimates. Gross floor area, net floor area, and usable area each serve different purposes in construction planning.
How to Calculate Area of Irregular Shapes
Irregular shapes cannot be calculated with a single formula. Use these approaches:
Decomposition method: Break the irregular shape into simpler shapes (rectangles, triangles, circles). Calculate each area separately and add them together.
Subtraction method: Draw a bounding rectangle around the shape, calculate its area, then subtract the areas of the regions outside the irregular shape.
Coordinate geometry (Shoelace formula): For a polygon with vertices at (x₁,y₁), (x₂,y₂), ..., (xₙ,yₙ), the area is: A = ½|Σ(xᵢ × yᵢ₊₁ - xᵢ₊₁ × yᵢ)| where the sum wraps around from the last vertex to the first.
Grid approximation: Overlay a grid on the shape and count the squares inside the boundary. This is less precise but useful for very complex shapes.
FAQ About Area Calculator
How do I calculate the area of a room?
Measure the length and width of the room in the same unit. Multiply them together. For a room that is 12 ft long and 10 ft wide: A = 12 × 10 = 120 ft². For rooms with alcoves or irregular shapes, break them into rectangles and add the areas.
How do I convert square feet to square meters?
Multiply the area in square feet by 0.092903. For example, 500 ft² × 0.092903 = 46.45 m². This calculator handles this conversion automatically in the unit conversions section.
How do I find the area of a circle?
Use the formula A = π × r², where r is the radius. If you know the diameter instead, divide it by 2 to get the radius, then apply the formula. A circle with radius 5 m has area π × 25 = 78.54 m².
How do I calculate the area of a triangle?
Use the formula A = ½ × base × height. The height must be the perpendicular distance from the base to the opposite vertex. A triangle with base 8 cm and height 6 cm has area ½ × 8 × 6 = 24 cm².
What is the difference between area and perimeter?
Area measures the space inside a shape (in square units), while perimeter measures the distance around the boundary (in linear units). A 4 × 4 square has area 16 m² and perimeter 16 m, but a 2 × 8 rectangle has area 16 m² and perimeter 20 m.
How do I calculate land area?
For rectangular land, multiply length × width. For irregular plots, use the decomposition method: divide the land into triangles and rectangles, calculate each, and sum them. Professional surveyors use GPS coordinates and the Shoelace formula for precise measurements.
What is the area of a sector of a circle?
A sector is a pie-shaped slice of a circle. Its area is A = ½ × r² × θ, where r is the radius and θ is the central angle in radians. For a 60° sector with radius 10 m: A = ½ × 100 × (60 × π/180) = ½ × 100 × 1.0472 = 52.36 m².
Why Use Our Area Calculator
This calculator is designed for accuracy, speed, and clarity. Every result includes a step-by-step breakdown so you can verify the math yourself. The visual diagrams help you confirm that you are entering the correct dimensions. With support for 9 area units and automatic conversions, you get the answer in whichever unit you need without opening a separate converter.
The tool runs entirely in your browser with no server calls, so your calculations are instant and private. Whether you are a student learning geometry, a contractor estimating materials, or a homeowner planning a renovation, this calculator gives you the answers you need without complexity.