Why Understanding Area Matters
Area is one of the most fundamental concepts in mathematics, and it shows up everywhere in daily life. Whether you are measuring a room for new flooring, calculating how much paint to buy for a wall, estimating the size of a garden plot, or comparing real estate listings, you need to know how to calculate area. Area tells you the amount of two-dimensional space a shape occupies, and it is always expressed in square units — square feet, square meters, square inches, and so on.
The concept is straightforward: area measures the surface enclosed within a boundary. A rectangle that is 10 feet long and 8 feet wide covers 80 square feet of space. But as shapes become more complex — triangles, circles, trapezoids, parallelograms, and irregular polygons — the formulas change. Each shape has its own area formula, and knowing the right one saves time and prevents costly mistakes in construction, design, and everyday planning.
In this guide, we cover the area formulas for every common shape, walk through step-by-step examples with real numbers, explain how to handle irregular shapes, and provide a complete area unit conversion table. If you want to skip the manual math, our free Area Calculator supports 11 different shapes with instant results, unit conversions, and a visual canvas preview.
Area Formulas for All Common Shapes
Below is a complete reference table of area formulas for geometry. Bookmark this page for quick access whenever you need to find area for any shape.
| Shape | Formula | Variables |
|---|---|---|
| Square | A = s² | s = side length |
| Rectangle | A = l × w | l = length, w = width |
| Triangle | A = ½ × b × h | b = base, h = height |
| Circle | A = π × r² | r = radius |
| Parallelogram | A = b × h | b = base, h = perpendicular height |
| Trapezoid | A = ½ × (a + b) × h | a, b = parallel sides, h = height |
| Rhombus | A = ½ × d1 × d2 | d1, d2 = diagonals |
| Ellipse | A = π × a × b | a = semi-major axis, b = semi-minor axis |
Every formula in this table is available in our Area Calculator, along with three additional shapes: regular pentagon, regular hexagon, and regular octagon. The calculator shows the formula used, performs unit conversions, and displays a visual preview of the shape on a canvas.
How to Calculate the Area of a Rectangle
The area of a rectangle is the simplest and most commonly used area calculation. You multiply the length by the width. This same formula applies to rooms, screens, tables, plots of land, and any four-sided shape with four right angles.
Area of Rectangle = length × width
Example: A room is 12 feet long and 10 feet wide.
Area = 12 × 10
Area = 120 square feetThis means you would need 120 square feet of flooring to cover the room. If tiles come in boxes of 20 square feet each, you would need 6 boxes (120 / 20 = 6). The area of a square is a special case of this formula where length and width are equal: A = s². A 9-foot by 9-foot room has an area of 81 square feet.
When dealing with dimensions in different units, convert them to the same unit first. If a wall is 3 meters wide and 280 centimeters tall, convert 280 cm to 2.8 meters before multiplying: 3 × 2.8 = 8.4 square meters. Our Height Converter can help with quick unit conversions between centimeters, meters, feet, and inches.
How to Calculate the Area of a Triangle
There are three main methods for calculating the area of a triangle, depending on what information you have available.
Method 1: Base and Height (Most Common)
When you know the base and the perpendicular height, use the standard formula:
Area of Triangle = ½ × base × height
Example: A triangle with a base of 14 cm and height of 9 cm.
Area = ½ × 14 × 9
Area = ½ × 126
Area = 63 square centimetersThe height must be perpendicular (at a 90-degree angle) to the base. If the triangle is obtuse, the height may fall outside the triangle, but the formula still works the same way.
Method 2: Heron's Formula (Three Sides Known)
When you know all three side lengths but not the height, use Heron's formula:
s = (a + b + c) / 2 (semi-perimeter)
Area = √(s(s-a)(s-b)(s-c))
Example: A triangle with sides of 7, 8, and 9 cm.
s = (7 + 8 + 9) / 2 = 12
Area = √(12 × 5 × 4 × 3)
Area = √720
Area = 26.83 square centimetersMethod 3: Coordinate Geometry
When you have the coordinates of all three vertices, use the coordinate formula:
Area = ½ |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
Example: Vertices at (1, 2), (4, 6), (7, 1).
Area = ½ |1(6 - 1) + 4(1 - 2) + 7(2 - 6)|
Area = ½ |5 + (-4) + (-28)|
Area = ½ |-27|
Area = 13.5 square unitsAll three methods give the same result for the same triangle. Choose whichever method matches the information you have. Our Area Calculator uses Method 1 by default but handles any triangle dimensions you provide.
How to Calculate the Area of a Circle
The area of a circle depends on a single measurement: the radius (the distance from the center to the edge). The formula uses pi (π), which is approximately 3.14159.
Area of Circle = π × r²
Example: A circular garden with a radius of 5 meters.
Area = π × 5²
Area = π × 25
Area = 3.14159 × 25
Area = 78.54 square metersIf you know the diameter instead of the radius, simply divide it by two. A pizza with a 16-inch diameter has a radius of 8 inches, giving an area of π × 64 = 201.06 square inches. This is why a single 16-inch pizza (201 sq in) has more area than two 10-inch pizzas (π × 25 × 2 = 157 sq in) — a fact that surprises many people.
For oval or egg-shaped areas, you need the area of an ellipse formula: A = π × a × b, where a and b are the semi-major and semi-minor axes. An elliptical pool that is 8 meters long and 5 meters wide has semi-axes of 4 and 2.5 meters, giving an area of π × 4 × 2.5 = 31.42 square meters.
How to Calculate the Area of a Trapezoid
A trapezoid (or trapezium in British English) has exactly two parallel sides. The area of a trapezoid formula averages the two parallel sides and multiplies by the height.
Area of Trapezoid = ½ × (a + b) × h
Where:
a = length of the first parallel side
b = length of the second parallel side
h = perpendicular height between the parallel sides
Example: A trapezoid with parallel sides of 8 m and 14 m,
and a height of 6 m.
Area = ½ × (8 + 14) × 6
Area = ½ × 22 × 6
Area = ½ × 132
Area = 66 square metersTrapezoids appear frequently in real-world structures: the cross-section of a drainage ditch, the shape of certain roof sections, decorative tiles, and oddly-shaped lots of land. The formula works regardless of whether the trapezoid is symmetric (isosceles) or has unequal non-parallel sides.
The area of a parallelogram is actually a simplified case of the trapezoid formula. Since both pairs of opposite sides are parallel and equal in a parallelogram, the formula reduces to A = base × height. A parallelogram with a base of 10 cm and a height of 7 cm has an area of 70 square centimeters.
Area of Irregular Shapes
Not every area you need to calculate will be a neat geometric shape. Rooms with alcoves, L-shaped gardens, oddly-shaped parcels of land, and custom floor plans all require a different approach. There are two practical methods for finding the area of irregular shapes.
The Decomposition Method
Break the irregular shape into regular shapes whose areas you can calculate, then add (or subtract) them together. This is the most accurate and most commonly used method.
Decomposition Method — Step by Step:
1. Identify regular shapes within the irregular outline
2. Measure dimensions for each sub-shape
3. Calculate the area of each sub-shape
4. Add areas together (subtract if shapes overlap)
Example: An L-shaped room
- Main section: 15 ft × 12 ft = 180 sq ft
- Extension: 8 ft × 6 ft = 48 sq ft
- Total area: 180 + 48 = 228 square feetFor shapes with curved sections, decompose the curved parts into semicircles or quarter-circles and apply the circle area formula to those sections. A window that is a rectangle topped by a semicircle can be split into a rectangular area plus half a circle's area.
The Grid Method
Place the irregular shape on graph paper or a grid overlay. Count the number of complete squares inside the shape, estimate partial squares (count each partial square as half), and multiply by the area represented by each grid square.
Grid Method — Step by Step:
1. Overlay the shape on a grid
2. Count full squares inside the boundary: e.g., 42
3. Count partial squares touching the boundary: e.g., 16
4. Estimate partial squares as half: 16 × 0.5 = 8
5. Total grid squares: 42 + 8 = 50
6. If each grid square = 1 cm², total area = 50 cm²The grid method is less precise than decomposition but works well for highly irregular shapes like lakes, leaf outlines, or organic land plots. Increasing the grid density (smaller squares) improves accuracy. For calculating what percentage of a larger area an irregular shape occupies, our Percentage Calculator can help with the proportional math.
Area Unit Conversions
One of the most common sources of errors in area measurement is unit conversion. Because area is two-dimensional, you must square the linear conversion factor. For instance, 1 foot = 12 inches, but 1 square foot = 144 square inches (12 × 12), not 12 square inches.
| From | To | Multiply By |
|---|---|---|
| Square inches (in²) | Square feet (ft²) | 0.006944 |
| Square feet (ft²) | Square inches (in²) | 144 |
| Square feet (ft²) | Square meters (m²) | 0.0929 |
| Square meters (m²) | Square feet (ft²) | 10.764 |
| Square yards (yd²) | Square meters (m²) | 0.8361 |
| Square meters (m²) | Square yards (yd²) | 1.196 |
| Acres | Square feet (ft²) | 43,560 |
| Acres | Square meters (m²) | 4,046.86 |
| Hectares (ha) | Acres | 2.471 |
| Square kilometers (km²) | Square miles (mi²) | 0.3861 |
| Square miles (mi²) | Acres | 640 |
| Hectares (ha) | Square meters (m²) | 10,000 |
When converting area units, always remember the squaring rule. If 1 meter = 3.281 feet, then 1 square meter = 3.281 × 3.281 = 10.764 square feet. Our Weight & Height Converter handles linear unit conversions between metric and imperial systems, which you can then square for area conversions.
Real-World Area Applications
Understanding how to find area has practical value in dozens of everyday situations. Here are the most common real-world applications of area calculations.
Flooring and Tiling
To determine how much flooring material you need, calculate the total floor area of the room. For a 14 × 11 foot room, the area is 154 square feet. If the flooring costs $4.50 per square foot, the material cost is 154 × $4.50 = $693. Always add 10% extra for waste and cuts: 154 × 1.10 = 169.4 square feet, so budget for 170 square feet.
Painting Walls
Calculate the area of each wall (height × width), then subtract the area of windows and doors. A wall that is 12 feet wide and 9 feet tall has 108 square feet. If it has a 3 × 5 foot window (15 sq ft), the paintable area is 93 square feet. One gallon of paint typically covers 350-400 square feet, so a single room with four walls often needs one gallon per coat.
Landscaping and Gardening
Mulch, sod, gravel, and topsoil are all sold by area coverage or volume. A circular flower bed with a 6-foot radius has an area of π × 36 = 113.1 square feet. If you need 3 inches of mulch, the volume is 113.1 × 0.25 feet = 28.3 cubic feet. For volume-based calculations, use our Volume Calculator to convert between cubic feet, cubic yards, and other units.
Real Estate
Property area is one of the most important factors in real estate valuation. A 1,500 square foot home priced at $300,000 costs $200 per square foot. Comparing price per square foot across listings helps you identify whether a property is fairly priced. Land is measured in acres (1 acre = 43,560 square feet) or hectares in countries using the metric system.
Frequently Asked Questions
What is the easiest way to calculate area?
The easiest way to calculate area depends on the shape. For rectangles and squares, simply multiply length by width. For triangles, multiply the base by the height and divide by two. For circles, multiply pi (3.14159) by the radius squared. If you are unsure of the shape or want instant results, use an online Area Calculator that handles all common shapes automatically.
How do you find the area of an irregular shape?
There are two main methods to find the area of an irregular shape. The decomposition method breaks the shape into regular shapes (rectangles, triangles, circles), calculates the area of each, and adds them together. The grid method places the shape on a grid, counts the full squares inside the shape, estimates partial squares as half, and multiplies by the area of each grid square. For complex irregular shapes, coordinate geometry or integration may be needed.
What is the difference between area and perimeter?
Area measures the amount of two-dimensional space inside a shape and is expressed in square units (such as square meters or square feet). Perimeter measures the total length of the boundary around a shape and is expressed in linear units (such as meters or feet). For example, a rectangle that is 5 meters by 3 meters has an area of 15 square meters and a perimeter of 16 meters.
How do I convert between different area units?
To convert between area units, use the square of the linear conversion factor. For example, since 1 foot equals 12 inches, 1 square foot equals 144 square inches (12 x 12). Common conversions include: 1 square meter = 10.764 square feet, 1 acre = 43,560 square feet, 1 hectare = 10,000 square meters, and 1 square mile = 640 acres. Always square the conversion factor when working with area.
What units are used to measure area?
Area is measured in square units. Common metric units include square millimeters (mm²), square centimeters (cm²), square meters (m²), hectares (ha), and square kilometers (km²). Common imperial units include square inches (in²), square feet (ft²), square yards (yd²), acres, and square miles (mi²). The choice of unit depends on the size of the area being measured — square centimeters for small objects, square meters for rooms, and acres or hectares for land.
Conclusion
Knowing how to calculate area is a practical skill that applies to construction, home improvement, gardening, real estate, and everyday problem solving. The formulas are straightforward once you know which shape you are working with: multiply length by width for rectangles, use ½ × base × height for triangles, apply π × r² for circles, and average the parallel sides times height for trapezoids.
For irregular shapes, the decomposition method (breaking complex shapes into simpler ones) is the most reliable approach. And when converting between area units, always remember to square the conversion factor — this is the single most common mistake in area measurement.
For instant, error-free calculations, use our free Area Calculator — it supports 11 shapes, displays the formula used, provides unit conversions, and keeps a calculation history. Pair it with our Volume Calculator for three-dimensional projects or our Percentage Calculator for proportional area comparisons. All tools are free, require no signup, and run entirely in your browser.