Area Calculator
Calculate the area of any shape instantly — rectangle, circle, triangle, trapezoid, ellipse, parallelogram and more — with step-by-step solutions.
📐 Area Calculator — Select a Shape
📐 Area Formulas Quick Reference
A = L × WA = a²A = π × r²A = ½ × b × hA = ½(a+b)×hA = b × hA = π × a × bA = (θ/360)×π×r²A = ½ × d₁ × d₂A = (3√3/2) × a²🔄 Unit Quick Conversions
What Is Area? A Complete Guide
Area is one of the most fundamental concepts in mathematics and geometry. It refers to the amount of two-dimensional space enclosed within the boundary of a flat shape or surface. In simpler terms, area tells you how much space a shape takes up on a flat plane.
Area is always expressed in square units — for example, square meters (m²), square feet (ft²), or square centimeters (cm²). The "square" in square units comes from the fact that area is computed by multiplying two linear measurements together, resulting in a two-dimensional measurement.
Understanding area is essential in virtually every professional and everyday context — from calculating how much paint you need for a wall, to determining the size of a land parcel, to designing floor plans, to solving geometry problems in school.
Area vs. Perimeter — What's the Difference?
These two concepts are often confused. Here is a simple way to remember the difference:
- Area — the space inside a shape (measured in square units: m², cm², ft²)
- Perimeter — the distance around the edge of a shape (measured in linear units: m, cm, ft)
For example, a 5 m × 4 m rectangle has an area of 20 m² but a perimeter of 18 m. You would use area to determine how much carpet you need for a room, and perimeter to determine how much skirting board to buy for the edges.
Why Is Area Important?
Area calculations are used across nearly every industry and walk of life:
- Construction and architecture: Calculating floor space, roof area, wall area for materials estimation
- Real estate: Measuring land plot sizes and property floor plans
- Agriculture: Determining field sizes for planting, irrigation, and yield estimation
- Interior design: Planning room layouts, flooring, and wall coverings
- Engineering: Computing cross-sectional areas for structural load calculations
- Science: Measuring surface areas for heat transfer, chemical reactions, and biological studies
- Art and crafts: Calculating fabric, canvas, or material requirements
Area Formulas for All Common Shapes
Each geometric shape has its own specific formula for calculating area. Below is a comprehensive reference covering all the major shapes, their formulas, and worked examples.
1. Rectangle
A rectangle has four sides and four right angles. Its opposite sides are equal in length.
A room measures 8 m × 5.5 m.
Area = 8 × 5.5 = 44 m²
You would need 44 m² of flooring to cover this room.
2. Square
A square is a special rectangle where all four sides are equal. Knowing just one side gives you the area.
A square tile has a side of 30 cm.
Area = 30² = 900 cm²
3. Circle
The area of a circle depends on its radius — the distance from the centre to the edge. Pi (π ≈ 3.14159) is a mathematical constant representing the ratio of any circle's circumference to its diameter.
A circular swimming pool has a radius of 4.5 m.
Area = π × 4.5² = π × 20.25 ≈ 63.62 m²
4. Triangle
The area of a triangle uses any base side and the perpendicular height from that base to the opposite vertex. Note that the height must be perpendicular (at 90°) to the base — it is not the slant length.
A triangular plot of land has a base of 24 m and a height of 15 m.
Area = ½ × 24 × 15 = 180 m²
5. Trapezoid (Trapezium)
A trapezoid has one pair of parallel sides (the two "bases") and the height is the perpendicular distance between those two sides.
Where a and b are the two parallel sides and h is the height.
A trapezoid has parallel sides of 8 m and 14 m, and a height of 6 m.
Area = ½ × (8 + 14) × 6 = ½ × 22 × 6 = 66 m²
6. Parallelogram
A parallelogram has two pairs of parallel sides. Its area formula is similar to a rectangle's, but the height must be perpendicular to the base — not the slant side.
A parallelogram has a base of 12 m and a perpendicular height of 7 m.
Area = 12 × 7 = 84 m²
7. Ellipse
An ellipse is an oval shape. It has two axes — the semi-major axis (a, the longer half) and the semi-minor axis (b, the shorter half).
An elliptical garden has a semi-major axis of 9 m and a semi-minor axis of 6 m.
Area = π × 9 × 6 = π × 54 ≈ 169.65 m²
8. Sector of a Circle
A sector is a "pie slice" portion of a circle, defined by a radius and a central angle.
A sector has a radius of 8 m and a central angle of 120°.
Area = (120/360) × π × 64 = ⅓ × π × 64 ≈ 67.02 m²
Complete Area Formula Reference Table
| Shape | Formula | Variables | Example (result) |
|---|---|---|---|
| Rectangle | A = L × W | L = length, W = width | 10 × 4 = 40 m² |
| Square | A = a² | a = side length | 7² = 49 m² |
| Circle | A = π × r² | r = radius | π × 5² ≈ 78.54 m² |
| Triangle | A = ½ × b × h | b = base, h = height | ½ × 12 × 9 = 54 m² |
| Trapezoid | A = ½(a+b)×h | a,b = parallel sides, h = height | ½(6+10)×5 = 40 m² |
| Parallelogram | A = b × h | b = base, h = perp. height | 8 × 6 = 48 m² |
| Ellipse | A = π × a × b | a = semi-major, b = semi-minor | π × 7 × 4 ≈ 87.96 m² |
| Sector | A = (θ/360)×π×r² | θ = angle (°), r = radius | (90/360)×π×36 ≈ 28.27 m² |
| Rhombus | A = ½ × d₁ × d₂ | d₁, d₂ = diagonals | ½ × 10 × 8 = 40 m² |
| Regular Hexagon | A = (3√3/2) × a² | a = side length | (3√3/2) × 6² ≈ 93.53 m² |
Area Unit Conversions — Complete Reference
Area measurements come in many different units depending on the scale and context of what you are measuring. Understanding how to convert between units is essential for accurate calculations.
Metric Area Units
The metric system is built on powers of 10, making conversions straightforward. Since area is two-dimensional, each step in the length scale corresponds to a 100× change in area (not 10×).
| Unit | Symbol | Equivalent | Typical Use |
|---|---|---|---|
| Square millimeter | mm² | 0.000001 m² | Small technical parts, precision work |
| Square centimeter | cm² | 0.0001 m² | Small surfaces, tiles, crafts |
| Square decimeter | dm² | 0.01 m² | Rarely used in practice |
| Square meter | m² | 1 m² | Rooms, floors, plots |
| Are | a | 100 m² | Small land areas |
| Hectare | ha | 10,000 m² | Agricultural land, large properties |
| Square kilometer | km² | 1,000,000 m² | Cities, countries, geographic regions |
Imperial / US Area Units
| Unit | Symbol | Equivalent | Metric Equivalent |
|---|---|---|---|
| Square inch | in² | — | 6.4516 cm² |
| Square foot | ft² | 144 in² | 0.0929 m² |
| Square yard | yd² | 9 ft² | 0.8361 m² |
| Acre | ac | 43,560 ft² | 4,046.86 m² |
| Square mile | mi² | 640 acres | 2.59 km² |
Quick Area Conversion Chart
How to Calculate Area of Irregular Shapes
Most real-world surfaces don't have perfect geometric shapes. A garden might be roughly L-shaped, a room might have a bay window alcove, or a land plot might have irregular boundaries. Here are the main methods for calculating irregular areas.
Method 1 — Decomposition into Regular Shapes
The most common approach is to divide an irregular shape into several regular shapes (rectangles, triangles, circles, etc.), calculate each area separately, then add or subtract them.
An L-shaped room can be split into two rectangles:
Rectangle 1: 6 m × 4 m = 24 m²
Rectangle 2: 3 m × 3 m = 9 m²
Total Area = 24 + 9 = 33 m²
Method 2 — Shoelace Formula (Coordinate Geometry)
For polygons where you know the coordinates of each vertex, the Shoelace Formula gives the exact area:
List the vertices in order, multiply across the diagonals of each pair, sum the results, and take half the absolute value. This method works for any polygon regardless of shape complexity.
Method 3 — Grid Counting
Overlay a grid of unit squares on the shape and count how many squares fall inside. Count full squares as 1, and partially covered squares as ½ each. This is an approximation method useful for hand-drawn shapes or maps.
Method 4 — Integration (Calculus)
For smoothly curved shapes described by mathematical functions, the area can be found using definite integrals. This is the basis for how computer software and GIS (geographic information systems) calculate areas of complex geographic regions.
Area Comparison: Same Perimeter, Different Shapes
One fascinating fact about area is that different shapes with the same perimeter can have very different areas. Among all shapes with a given perimeter, a circle always has the greatest area — this is known as the isoperimetric inequality.
Perimeter fixed at 40 units for all shapes
Real-World Area Calculation Examples
Let's look at practical, everyday scenarios where area calculations are essential.
Example 1 — Flooring a Room
You want to install hardwood flooring in your living room. The room measures 7.2 m × 4.8 m, with a rectangular fireplace alcove of 1.2 m × 0.9 m that does not need flooring.
- Room area = 7.2 × 4.8 = 34.56 m²
- Alcove area = 1.2 × 0.9 = 1.08 m²
- Net floor area = 34.56 − 1.08 = 33.48 m²
- Adding 10% wastage for cuts: 33.48 × 1.10 = 36.83 m² to order
Example 2 — Painting a Wall
A wall is 4.5 m wide and 2.8 m tall, with two windows each measuring 1.2 m × 1.0 m.
- Wall area = 4.5 × 2.8 = 12.6 m²
- Window areas = 2 × (1.2 × 1.0) = 2.4 m²
- Paintable area = 12.6 − 2.4 = 10.2 m²
- If paint covers 10 m² per litre: need approximately 1.02 litres per coat
Example 3 — Circular Garden Bed
You want to lay mulch in a circular garden bed with a diameter of 3.6 m. Mulch costs $12 per m².
- Radius = 3.6 / 2 = 1.8 m
- Area = π × 1.8² = π × 3.24 ≈ 10.18 m²
- Cost = 10.18 × $12 = $122.15
Example 4 — Agricultural Field
A trapezoidal field has parallel sides of 180 m and 240 m, with a perpendicular distance of 120 m between them.
- Area = ½ × (180 + 240) × 120 = ½ × 420 × 120 = 25,200 m²
- In hectares: 25,200 / 10,000 = 2.52 hectares
- In acres: 25,200 / 4,046.86 ≈ 6.22 acres
Area in Context: Common Size References
Advanced Area Concepts
Heron's Formula — Triangle Area from Three Sides
If you know all three side lengths of a triangle but not the height, you can use Heron's Formula. First compute the semi-perimeter, then apply the formula.
s = (5 + 7 + 8) / 2 = 10
A = √(10 × 5 × 3 × 2) = √300 ≈ 17.32 m²
Surface Area vs. Flat Area
When dealing with three-dimensional objects, we often talk about surface area — the total area of all faces of a solid. This is distinct from the flat (2D) area we've been discussing. For example:
| 3D Shape | Surface Area Formula | Volume Formula |
|---|---|---|
| Cube (side a) | 6a² | a³ |
| Rectangular box | 2(lw + lh + wh) | l × w × h |
| Sphere (radius r) | 4πr² | (4/3)πr³ |
| Cylinder (r, h) | 2πr(r + h) | πr²h |
| Cone (r, l, h) | πr(r + l) | (1/3)πr²h |
Scale Factor and Area
When a shape is scaled by a factor of k in both dimensions, its area changes by a factor of k². This is a critical concept in maps, blueprints, and engineering drawings.
Integration Method for Curved Areas
For areas bounded by curves, calculus provides the definite integral method. The area under a curve y = f(x) between x = a and x = b is:
This underpins how digital tools, GPS systems, and geographic software calculate areas of irregular real-world regions — from field boundaries to national park extents.