Area Calculator - Free Online Tool

Calculate the area of any shape with formulas, step-by-step workings and unit conversion

Calculating the area of a geometric shape is one of the most practical mathematical skills in everyday life. Whether you are planning a home renovation, estimating how much paint you need, laying down flooring, designing a garden, or working through a geometry homework problem, knowing how to find the area of different shapes quickly and accurately makes a real difference. Our free area calculator supports 19 common shapes and gives you the answer in seconds — along with the formula used, a step-by-step breakdown, and automatic conversion to eight different units. Area is a measure of the two-dimensional space enclosed within a boundary. It answers the question 'how much surface does this shape cover?' and is expressed in squared units — square metres (m²), square feet (ft²), square centimetres (cm²), and so on. While simple shapes like squares and rectangles need only a single multiplication, more complex shapes like triangles, circles, and polygons require specific formulas and sometimes multiple inputs. Our calculator handles all of these automatically so you never have to worry about which formula to use. Our tool covers all the shapes you are likely to encounter in real life. The basic set — square, rectangle, triangle, and circle — handles the vast majority of everyday calculations. The quadrilateral group adds parallelogram, trapezoid, rhombus, kite, and irregular quadrilateral for more complex land and floor plans. The circular group covers ellipses, sectors, semicircles, circle segments, and annuli (rings). The polygon group supports named shapes including pentagon, hexagon, and octagon, plus any regular n-sided polygon. Finally, the special shapes group includes the rectangular border, which is perfect for calculating the area of frames, borders, and hollow rectangular regions. Triangle calculations deserve special mention. A triangle's area depends on what measurements you know, and our calculator supports six different solve methods: base and height (the classic formula), SSS using Heron's formula when you know all three sides, SAS when you have two sides and the included angle, ASA when you know two angles and the side between them, AAS when you have two angles and a non-included side, and HL for right triangles with the hypotenuse and one leg. All angle-based methods accept both degrees and radians. Unit flexibility is built in from the start. You can work in millimetres, centimetres, metres, kilometres, inches, feet, yards, or miles. The result is automatically shown in your chosen squared unit, and a conversion table below the main result displays the equivalent area in all eight units simultaneously — saving you extra conversion steps. For each shape, you get a labeled SVG diagram showing which dimension goes where, so there is no ambiguity about what each input field means. The formula card displays the symbolic formula used and a substituted step-by-step calculation so you can verify the working or learn from it. A reference chart shows how your calculated area compares to the size of a football field (7,140 m²), adding helpful real-world context to otherwise abstract numbers.

Understanding Area Calculation

¿Qué es el Área?

Area is the measure of two-dimensional space enclosed within a closed boundary. It tells you how much surface a shape covers. Area is always expressed in squared units — for example, square metres (m²), square feet (ft²), or square centimetres (cm²). This is because area involves two perpendicular dimensions multiplied together. For a 3 m by 4 m rectangle, the area is 12 m², meaning the rectangle covers 12 individual 1 m × 1 m squares. Understanding area is fundamental to architecture, construction, agriculture, interior design, and many other fields. In everyday life, area calculations tell you how much paint to buy for a wall, how many floor tiles you need, how large a piece of land is, or how much fabric is required for a sewing project.

How Is Area Calculated?

Each geometric shape has its own area formula derived from its defining properties. The simplest — a square — just squares one side (A = s²). A rectangle multiplies length by width (A = l × w). A triangle's basic formula is half the base times the perpendicular height (A = ½ × b × h), but Heron's formula can compute it from three side lengths, and trigonometric methods (SAS, ASA, AAS) use angles and sides together. A circle's area comes from the famous formula A = π × r², where r is the radius and π ≈ 3.14159. An ellipse generalises this to A = π × a × b using its two semi-axes. Irregular shapes like trapezoids, annuli, and regular polygons each have their own precise formulas. This calculator implements all of them and converts the raw formula result into your chosen unit of area automatically.

Why Accurate Area Calculation Matters

Errors in area calculation can be expensive. Underestimating the area of a room means buying too little flooring or paint, requiring a second trip to the store. Overestimating wastes materials and money. In agriculture, accurate field area determines how much seed, fertiliser, or irrigation water is needed — getting this wrong affects both costs and crop yields. In construction, precise area measurements underpin structural load calculations, material estimates, and cost projections. Even in everyday contexts like garden planning or tiling a bathroom, being off by 10% on area can mean significant extra cost or wasted materials. Our calculator eliminates arithmetic errors by handling every formula precisely, and the unit conversion table helps you work confidently across metric and imperial systems.

Limitaciones y advertencias

This calculator computes flat (2D) area only. It does not account for surface curvature — for curved 3D surfaces like spheres or cylinders, a surface area calculator is needed. For triangles, the SSS method (Heron's formula) requires valid triangle inequality: each side must be shorter than the sum of the other two. The HL method applies only to right triangles and requires the hypotenuse to be strictly longer than the given leg. For the sector and circle-segment shapes, angles must be positive and less than 360° (or 2π radians). For the rectangular border shape, the inner dimensions must be strictly smaller than the outer dimensions. The annulus (ring) requires the inner radius to be smaller than the outer radius. All inputs must be positive numbers; the calculator will show no result if any required value is missing or invalid.

Area Formulas Reference

Rectángulo

A = L × W

The area of a rectangle is the product of its length and width. This is the most commonly used area formula for rooms, walls, floors, and rectangular plots of land.

Circle

A = πr²

The area of a circle equals pi (≈ 3.14159) multiplied by the square of the radius. For a circle with diameter d, use r = d/2.

Triangle (Base and Height)

A = ½ × b × h

The area of a triangle is one-half the base multiplied by the perpendicular height. The height must be measured at a right angle to the chosen base.

Trapezoid

A = ½ × (a + b) × h

The area of a trapezoid is half the sum of the two parallel sides (bases a and b) multiplied by the perpendicular height between them.

Ellipse

A = π × a × b

The area of an ellipse is pi multiplied by the semi-major axis (a) and the semi-minor axis (b). When a = b, this reduces to the circle formula.

Area Reference Tables

Area Formulas for Common Shapes

Quick reference of area formulas for the most frequently used geometric shapes, with the variables defined for each.

Forma	Fórmula	Variables
Square	A = s²	s = side length
Rectángulo	A = L × W	L = length, W = width
Triangle	A = ½bh	b = base, h = height
Circle	A = πr²	r = radius
Parallelogram	A = b × h	b = base, h = perpendicular height
Trapezoid	A = ½(a + b) × h	a, b = parallel sides, h = height
Ellipse	A = πab	a = semi-major, b = semi-minor
Rhombus	A = ½ × d₁ × d₂	d₁, d₂ = diagonals
Regular Polygon	A = (ns²) / (4tan(π/n))	n = sides, s = side length
Annulus	A = π(R² − r²)	R = outer radius, r = inner radius

Area Unit Conversion Factors

Multiply an area in the left-column unit by the factor to convert to the top-row unit.

From	To m²	To ft²	To acres
1 cm²	0.0001	0.00107639	—
1 m²	1	10.7639	0.000247105
1 km²	1,000,000	10,763,910	247.105
1 in²	0.000645	0.00694444	—
1 ft²	0.092903	1	0.0000229568
1 yd²	0.836127	9	0.000206612
1 acre	4,046.86	43,560	1

Worked Examples

Area of an L-Shaped Room (Decomposition Method)

A room has an L-shape: the main section is 5 m × 4 m and the extension is 3 m × 2 m.

Decompose the L-shape into two rectangles

Calculate the main section: 5 × 4 = 20 m²

Calculate the extension: 3 × 2 = 6 m²

Add the two areas: 20 + 6 = 26 m²

The total area of the L-shaped room is 26 m².

Area of a Circular Garden with 15 ft Radius

A circular garden has a radius of 15 feet. Find its area in square feet and square metres.

Apply the circle formula: A = πr²

Substitute r = 15: A = π × 15² = π × 225

Calculate: A ≈ 3.14159 × 225 ≈ 706.86 ft²

Convert to m²: 706.86 × 0.092903 ≈ 65.67 m²

The garden area is approximately 706.86 ft² (65.67 m²).

Area of a Trapezoid-Shaped Plot

A land plot has parallel sides of 30 m and 20 m, with a perpendicular distance of 15 m between them.

Apply the trapezoid formula: A = ½(a + b) × h

Substitute values: A = ½ × (30 + 20) × 15

Simplify: A = ½ × 50 × 15 = 375 m²

Convert to acres: 375 × 0.000247105 ≈ 0.0927 acres

The plot area is 375 m² (approximately 0.093 acres).

How to Use the Area Calculator

Select a Shape

Click one of the 19 shape buttons grouped by type — Basic, Quadrilaterals, Circular, Polygons, or Special. The diagram and input fields will update instantly to match your chosen shape.

Choose Your Unit and Enter Dimensions

Select your preferred length unit (mm, cm, m, km, in, ft, yd, or mi) from the unit buttons. Then enter the required dimensions in the input fields. For triangles, first choose your preferred solve method (Base+Height, SSS, SAS, ASA, AAS, or HL).

Revisar el resultado

The area appears instantly in your chosen squared unit along with the perimeter (where applicable). The formula card shows the symbolic formula and a substituted step-by-step calculation. A unit conversion table displays the area in all eight units simultaneously.

Copy or Use the Result

Click the Copy button to copy the area value to your clipboard. The real-world reference chart below the results shows how your shape's area compares to a standard football field, giving helpful scale context for the number.

Preguntas Frecuentes

How do I calculate the area of a triangle when I only know all three sides?

Use Heron's formula. First compute the semi-perimeter: s = (a + b + c) / 2. Then the area is A = √(s(s−a)(s−b)(s−c)). For example, for a triangle with sides 5, 6, and 7: s = (5+6+7)/2 = 9. Area = √(9 × 4 × 3 × 2) = √216 ≈ 14.70 square units. In this calculator, select Triangle, choose the SSS method, and enter the three side lengths — the result is computed automatically. Remember that for a valid triangle, each side must be shorter than the sum of the other two sides.

What is the difference between a sector and a circle segment?

A sector is the 'pie slice' shape bounded by two radii and an arc — like a slice of pizza. Its area is A = (θ/360) × π × r² in degrees, or A = ½r²θ in radians. A circle segment is the region between a chord and the arc it cuts off — like the curved piece left after slicing a chord across a circle. Its area is A = ½r²(θ − sinθ), where θ is the central angle in radians. The segment is always smaller than or equal to the corresponding sector for the same central angle, because the triangular portion between the two radii and the chord is subtracted.

How do I find the area of an annulus (ring shape)?

An annulus is the region between two concentric circles — like a washer or a circular border. Its area is simply the area of the outer circle minus the area of the inner circle: A = π × (R² − r²), where R is the outer radius and r is the inner radius. For example, if R = 8 cm and r = 5 cm, then A = π × (64 − 25) = π × 39 ≈ 122.52 cm². The inner radius must always be smaller than the outer radius. In this calculator, select Annulus, enter R and r in your chosen unit, and the area is calculated instantly.

Why does the area change when I switch units?

Unit switching does not change the physical size of the shape — it changes how you describe the dimensions you enter. If you type '5' in the side field with 'ft' selected, the calculator treats that as 5 feet. If you then switch to 'm', it treats the same '5' as 5 metres, which is a physically much larger shape (about 16.4 feet). Always set your unit before entering values. The unit conversion table in the results shows your calculated area converted into all other units, which is useful when you want the same result expressed in a different unit.

What is a regular polygon and how is its area calculated?

A regular polygon is a flat shape with n equal sides and equal interior angles. Common examples include the equilateral triangle (n=3), square (n=4), pentagon (n=5), hexagon (n=6), and octagon (n=8). The general area formula is A = (n × s²) / (4 × tan(π/n)), where n is the number of sides and s is the side length. For a hexagon with side 5 m: A = (6 × 25) / (4 × tan(30°)) = 150 / (4 × 0.5774) ≈ 64.95 m². This calculator supports any regular polygon — just select Regular Polygon, enter the number of sides, and the side length.

What is the rectangular border shape and when would I use it?

The rectangular border shape calculates the area of a frame or hollow rectangular region — the area between a larger outer rectangle and a smaller inner rectangle. The formula is A = (L_outer × W_outer) − (L_inner × W_inner). This is useful for picture frames, window borders, pool surrounds, or path borders around rectangular lawns. For example, a frame with outer dimensions 60 cm × 40 cm and inner dimensions 50 cm × 30 cm has a border area of (60×40) − (50×30) = 2400 − 1500 = 900 cm². The inner dimensions must be smaller than the outer ones.