Calculate the Area of 2D Shapes
Area is the amount of two-dimensional space enclosed by a shape. It is measured in square units, such as square meters (m²), square centimeters (cm²), square feet (ft²), square inches (in²), acres, and other area units.
This area calculator lets you calculate the area of common geometric shapes by entering the dimensions you already know. Depending on the shape, you can also see related measurements such as perimeter, circumference, diagonals, apothem, unit conversions, and material requirements.
Instead of remembering a different formula for every shape, choose the geometry you need, enter its measurements, and check the calculation. The tool supports standard shapes as well as more specialized cases such as Heron's formula triangles, circular sectors, annuli, regular polygons, and irregular polygons defined by coordinates.
For simple shapes, the calculation is usually direct. For more complex shapes, the calculator applies the corresponding geometric formula and shows the result in the selected unit system.
Use these links to jump directly to the mathematical method you need:
What Does Area Mean?
Area describes how much surface a two-dimensional figure covers.
A rectangle measuring 4 meters by 5 meters covers:
The answer is expressed in square meters because the dimensions are measured in meters and two dimensions are multiplied together.
This distinction matters:
- Length is one-dimensional.
- Perimeter measures the boundary of a shape.
- Area measures the enclosed surface.
- Volume measures three-dimensional space (explored further in our Volume Calculator).
For example, a room may have a floor area of 20 m² while its perimeter is measured in linear meters. Those are fundamentally different physical quantities and should not be confused.
Rectangle Area
A rectangle is one of the simplest shapes for calculating area because its opposite sides are equal and its angles are right angles.
where A = area, l = length, and w = width.
The same basic relationship applies to a square. Because all sides of a square are equal:
Suppose a rectangular floor is Length = 8 m, Width = 5 m.
A = 8 × 5 = 40 m²
So the floor covers 40 square meters.
The rectangle and related elementary area formulas are standard geometry results documented by OpenStax.
Rectangle area is useful for floors, walls, rooms, gardens, sheets of material, rectangular lots, panels, tiles, and boards. When estimating materials, the geometric area may need to be increased to account for cutting waste, unusable sections, or installation losses.
Triangle Area
The most common triangle area formula uses the base and its perpendicular height. For solving complete triangles with side angles and trigonometric relationships, visit our dedicated Triangle Area Calculator.
where b = base and h = perpendicular height.
For a triangle with Base = 10 m and Height = 6 m:
A = ½ × 10 × 6 = 30 m²
A triangle with a given base and perpendicular height occupies half the area of a parallelogram with the same base and height. That is the geometric reason for the factor of ½.
Important: height is not necessarily a sideFor an angled triangle, the height is the perpendicular distance from the base to the opposite vertex. Using a sloping side as the height produces an incorrect area.
Triangle Area When All Three Sides Are Known (Heron's Formula)
When the three side lengths are known but the height is not, Heron's formula can be used. First calculate the semiperimeter:
Then evaluate:
For example, for sides 7, 8, and 9: s = (7 + 8 + 9)/2 = 12, so A = √[12(12-7)(12-8)(12-9)] = √720 ≈ 26.8328 square units.
Circle Area
For a circle, area depends on the radius.
where A = area, r = radius, and π ≈ 3.14159265359.
For a circle with radius 5 m:
A = π × (5)² = 25π ≈ 78.5398 m²
The corresponding circumference is C = 2πr, so for r = 5 m, C ≈ 31.4159 m.
A frequent mistake is entering the diameter as though it were the radius. If diameter is d, then r = d / 2 and A = π(d/2)². Do not substitute the full diameter into πr², which would overestimate area by a factor of 4.
Sector and Annulus Area
A circular sector is a portion of a circle bounded by two radii and an arc. For an angle measured in degrees:
The corresponding arc length is L = (θ / 360°) × (2πr).
For r = 10 m and θ = 90°: A = (90/360) × π(10)² = 25π ≈ 78.5398 m², and arc length L = 5π ≈ 15.7080 m.
Annulus Area (Concentric Ring)
An annulus is the ring-shaped region between two concentric circles. Its area is:
where R = outer radius and r = inner radius. The inner radius must be smaller than the outer radius (0 < r < R). For R = 10 m and r = 5 m: A = π(10² - 5²) = 75π ≈ 235.6194 m².
Trapezoid and Parallelogram Area
Trapezoid
A trapezoid has two parallel bases. If the bases are b₁ and b₂, and perpendicular height is h:
For example, b₁ = 10, b₂ = 6, h = 4 yields A = ½(10 + 6)(4) = 32 square units. Do not use the slanted side as height; height is strictly the perpendicular distance between the parallel bases.
Parallelogram
The area of a parallelogram is:
For b = 10 and h = 6, A = 10 × 6 = 60 square units. Even when tilted, area depends on base and perpendicular height rather than the sloping side.
Rhombus and Kite Area
For a rhombus or kite, a convenient formula uses its perpendicular diagonals:
For example, if diagonals are d₁ = 10 and d₂ = 8, then A = (10 × 8) / 2 = 40 square units. Be careful to use actual diagonal lengths rather than arbitrary side lengths.
Regular Polygon Area
A regular polygon has equal side lengths and equal interior angles (equilateral triangles, squares, regular pentagons, hexagons, octagons). For a regular polygon, area can be calculated with the apothem and perimeter:
where a = apothem (inradius perpendicular from center to side midpoint), and p = n × s is perimeter.
For n = 6 sides with side length s = 5 m:
Perimeter p = 6 × 5 = 30 m.
Apothem a = 5 / [2 tan(π/6)] ≈ 4.3301 m.
A = ½ × 4.3301 × 30 ≈ 64.9519 m²
Irregular Polygon Area (Shoelace Formula)
Irregular polygons do not have one simple formula based only on a small number of side lengths. When the vertices are known as Cartesian coordinates, the Shoelace Formula provides a systematic way to calculate the area of a simple polygon.
with the last vertex connected back to the first vertex. This method is documented by Wolfram MathWorld as Gauss's area formula or the surveyor's formula.
For vertices (0,0), (10,0), (10,6), (4,10), and (0,6), the calculator evaluates the signed coordinate determinants directly to return an exact area of 80 square units and perimeter of approximately 34.87 units.
Vertices should describe the boundary in sequence (either clockwise or counterclockwise) and form a non-self-intersecting simple polygon.
How to Choose the Correct Area Formula
The easiest way to select a formula is to start with the geometric information you actually know:
| Shape | Typical Inputs | Area Formula |
|---|---|---|
| Rectangle | length, width | A = l × w |
| Square | side (s) | A = s² |
| Triangle | base, perpendicular height | A = ½ × b × h |
| Triangle (3 sides) | sides a, b, c | Heron: √[s(s-a)(s-b)(s-c)] |
| Circle | radius (r) | A = π × r² |
| Circular Sector | radius, central angle θ | A = (θ / 360°) × πr² |
| Annulus (Ring) | outer R, inner r | A = π(R² - r²) |
| Trapezoid | bases b₁, b₂, height h | A = ½(b₁ + b₂)h |
| Parallelogram | base b, height h | A = b × h |
| Rhombus / Kite | diagonals d₁, d₂ | A = ½ × d₁ × d₂ |
| Regular Polygon | side count n, side s / apothem a | A = ½ × a × p |
| Irregular Polygon | Cartesian vertices (x, y) | Shoelace: ½|∑(x_i y_{i+1} - x_{i+1} y_i)| |
Area Units and Conversions
Area units are squared because area measures two dimensions:
Similarly, 1 ft² is an area unit (144 in²), not a linear foot measurement. Key international standards (NIST SP 811) include:
- 1 m² ≈ 10.7639 ft²
- 1 ft² = 0.09290304 m²
- 1 acre = 43,560 ft² = 4,046.8564 m²
- 1 hectare (ha) = 10,000 m² ≈ 2.47105 acres
- 1 square mile (sq mi) = 640 acres ≈ 2,589,988.11 m²
Why you should not convert dimensions incorrectly: If a rectangle is 2 m × 3 m, its area is 6 m². To express this in square feet, multiply by the area factor (6 × 10.7639 ≈ 64.58 ft²). Do not convert 2 m to feet and then multiply by an area factor again, which would double-count the conversion.
Worked Examples
Length = 4.5 m, Width = 3.2 m.
A = 4.5 × 3.2 = 14.4 m²
The room has 14.4 m² of floor area.
Radius = 4 m.
A = π × (4)² = 16π ≈ 50.2655 m²
Approximately 50.27 m² of ground is enclosed.
Base = 12 ft, Height = 7 ft.
A = ½ × 12 × 7 = 42 ft²
Enclosed panel area is 42 square feet.
Parallel sides = 12 m and 8 m, Height = 5 m.
A = ½(12 + 8)(5) = 50 m²
Enclosed trapezoidal surface is 50 square meters.
Side = 5 m, Perimeter = 30 m, Apothem ≈ 4.3301 m.
A = ½ × 4.3301 × 30 ≈ 64.9519 m²
Hexagon area is approximately 64.95 m².
Points (0,0), (10,0), (10,6), (4,10), (0,6).
Shoelace Area = 80 m² (P ≈ 34.87 m)
Evaluates signed vertex determinant sums directly.
Area vs Perimeter: What Is the Difference?
Area and perimeter answer fundamentally different questions:
How much surface does the shape cover?
Measured in 2D squared units (m², ft², in²).
10m × 5m → A = 50 m²
How long is the complete boundary?
Measured in 1D linear units (m, ft, in).
2(10 + 5) → P = 30 m
Using Area for Material Estimates
Geometric area is often the first step in estimating material quantities. Suppose a floor has an area of 50 m² and you anticipate a 10% cutting waste margin:
At a unit cost of $20 per square meter, the total estimated cost is 55 × $20 = $1,100. This planning estimate provides transparency before purchasing materials.
Common Area-Calculation Mistakes
For triangles, parallelograms, and trapezoids, height strictly denotes the perpendicular distance, never an arbitrary sloping edge.
Substituting diameter into πr² produces an area four times too large because the radius is squared.
Calculating length in meters and width in feet without converting to a common unit causes substantial mathematical error.
Linear unit conversions cannot be directly applied to area results without squaring the conversion ratio.
Regular polygons use side count and apothem, whereas irregular plots require Cartesian coordinates and the Shoelace formula.
An annulus requires r < R, triangle sides must satisfy the triangle inequality, and polygons require n ≥ 3. The calculator explicitly rejects invalid inputs rather than silently clamping.
Core Formulas at a Glance
Frequently Asked Questions
References and Methodology
For additional mathematical background and verification standards, consult the following authoritative references:
- OpenStax — Geometric Formulas: Describes standard elementary area relationships for rectangles, triangles, circles, parallelograms, trapezoids, rhombi, and regular polygons.
- Wolfram MathWorld — Shoelace Formula: Authoritative reference for the coordinate-based polygon area formula, also known as Gauss's area formula or the surveyor's formula.
- NIST (National Institute of Standards and Technology) — Special Publication 811: Official conversion factors for U.S. customary measures and SI metric area units (square meters, square feet, square yards, acres, and hectares).