Area Calculator - Free Online Tool

Calculate area for 19 shapes with formulas, perimeter, and unit conversions

The Area Calculator is an essential geometry tool for students, engineers, architects, landscapers, DIY homeowners, and anyone who needs to measure two-dimensional space accurately. Whether you are planning a room renovation, calculating how much paint you need, sizing a garden bed, estimating roofing materials, or working through a geometry homework problem, knowing the area of a shape is the fundamental first step. Area is defined as the amount of two-dimensional space enclosed within the boundary of a shape. It is always measured in squared units — square meters (m²), square feet (ft²), square inches (in²), and so on. Different shapes have different formulas because their boundaries behave differently. A rectangle multiplies two perpendicular sides; a circle uses the radius squared multiplied by pi; a triangle uses half the base times height. Understanding which formula applies — and correctly substituting your measurements — is where errors most commonly occur. This calculator eliminates those errors by letting you select your shape, enter your measurements in any unit, and instantly see the result. It supports 19 different shapes ranging from simple squares and rectangles to complex shapes like annuli (rings), circle segments, and rectangular borders. For triangles alone, six different solving methods are supported: base and height, SSS (Heron's formula using all three sides), SAS (two sides and included angle), ASA, AAS, and hypotenuse-leg for right triangles. Practical applications are everywhere. Flooring installers calculate room area to order the correct amount of tile or carpet. Painters estimate wall or ceiling area to determine how many gallons of paint to buy. Landscapers measure garden beds and lawns to calculate turf, seed, or mulch quantities. Farmers assess field area to plan irrigation and planting. Architects and structural engineers compute cross-sectional areas for load calculations. Students use area formulas daily in geometry, trigonometry, and calculus courses. The tool also provides perimeter (or circumference) alongside the area, since both measurements are often needed together. Building a fence? You need the perimeter. Tiling a floor? You need the area. Painting a wall with a border? You need both. Having both results from a single calculation saves time and reduces errors from switching between multiple calculators. Unit flexibility is another key feature. Measurements might come in feet and inches on a blueprint, meters from a laser distance tool, or centimeters from a tape measure. All inputs use the same unit selector — choose from millimeters, centimeters, meters, kilometers, inches, feet, yards, or miles — and the result is displayed in the corresponding squared unit. A conversion table beneath the primary result also expresses the area in all common units simultaneously, making it easy to cross-check values or share results with colleagues using different measurement systems. For shapes with angles — sectors, and several triangle methods — the angle can be entered in either degrees or radians, whichever is more natural for your workflow. A toggle switch between the two modes appears automatically when an angle input is required. The formula card displayed with every result shows the exact formula used, with your values substituted in, so you can verify the calculation and understand the mathematics behind it. This is especially useful for students who need to show their working, and for professionals who need to document their calculations for compliance or quality control purposes.

Understanding Area Calculations

面積とは何ですか？

Area is the measure of the two-dimensional space enclosed within a shape's boundary. It quantifies how much flat surface a shape covers. Area is always expressed in squared units because it represents length multiplied by length in two dimensions — for example, a square measuring 4 meters on each side has an area of 4 × 4 = 16 square meters (m²). Unlike length (one-dimensional) or volume (three-dimensional), area describes surfaces: floors, walls, fields, screens, and cross-sections. Every discipline that involves physical surfaces — architecture, civil engineering, agriculture, interior design, manufacturing, and everyday home improvement — depends on accurate area calculations. Without knowing area, it is impossible to correctly estimate material quantities, costs, or capacities.

How Is Area Calculated?

Each shape has its own formula derived from its geometric properties. Simple shapes like squares (A = s²) and rectangles (A = l × w) multiply perpendicular sides. Triangles (A = ½bh) use half the product of base and perpendicular height because a triangle is half a parallelogram. Circles (A = πr²) use the radius squared times pi, reflecting the relationship between radius and arc length. Ellipses extend this to A = πab using both semi-axes. Trapezoids average the two parallel bases before multiplying by height: A = ½(b₁ + b₂)h. Complex shapes like regular polygons use trigonometric relationships — A = (ns²) / (4 tan(π/n)) — while Heron's formula for SSS triangles uses the semi-perimeter s = (a+b+c)/2 to compute A = √(s(s−a)(s−b)(s−c)) without needing height explicitly. For shapes with rounded elements like sectors (A = θr²/2) and annuli (A = π(R²−r²)), the formulas incorporate pi and radius relationships.

Why Does Area Matter?

Accurate area measurement directly affects costs, quantities, and safety. When ordering flooring, overestimating wastes money while underestimating means a second purchase trip and potential color lot mismatches. In agriculture, field area determines seed rates, fertilizer quantities, and irrigation system sizing — errors translate into poor yields or wasted inputs. For structural engineers, cross-sectional area of a beam or column directly determines its load-bearing capacity; underestimating this is a safety risk. In roofing, area calculation must account for pitch and overlap to avoid material shortages. For painters, one gallon of paint covers approximately 350–400 square feet; knowing area precisely means ordering the right amount. Even in digital contexts, area matters: screen resolution, sensor size, and print media all involve area calculations that affect quality and cost.

制限と注意事項

This calculator computes flat (planar) 2D area only. It does not account for surface curvature, slope, or three-dimensional complexity. A sloped roof, for example, has more area than its horizontal footprint suggests — you would need to factor in pitch angle separately. Similarly, for irregular shapes not covered by the supported set, the calculator cannot be used directly; instead, decompose the shape into supported sub-shapes and sum their areas. Measurement accuracy is also critical: a 5% error in a dimension causes approximately a 10% error in area (since area scales as the square). For very large or very small measurements, floating-point precision limits in JavaScript mean results beyond about 15 significant digits may be rounded. Always verify critical calculations with physical measurements and professional review when safety or large expenditures are involved.

Area Formulas Reference

長方形

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.

Shape	式	変数
Square	A = s²	s = side length
長方形	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 Your Shape

Use the Shape dropdown to choose from 19 supported shapes grouped into Basic, Quadrilaterals, Circular, and Polygons. For triangles, an additional Method selector appears letting you choose Base & Height, SSS (Heron's), SAS, ASA, AAS, or Hypotenuse & Leg.

Choose Your Unit

Select the unit of measurement from the Unit dropdown — mm, cm, m, km, in, ft, yd, or mi. All dimension inputs use this single unit, so make sure all your measurements are in the same unit before entering them.

Enter Dimensions

Fill in the dimension fields that appear for your selected shape. The form updates dynamically — only the relevant inputs for your shape are shown. For shapes requiring an angle (Sector, SAS triangle, etc.), use the Angle Unit toggle to switch between degrees and radians.

結果を読む

Results appear instantly as you type. The area is shown prominently at the top, followed by the perimeter or circumference, the substituted formula showing your exact values, a table of area conversions in all common units, and a visual shape diagram with labeled dimensions.

よくある質問

How do I calculate the area of an irregular shape?

For irregular shapes not directly supported by this calculator, the standard approach is shape decomposition — break the irregular shape into smaller, simpler shapes (rectangles, triangles, circles) whose areas you can calculate individually, then add them all together. For example, an L-shaped room can be split into two rectangles. A half-circle window can be added to a rectangular wall area. The Multiple Shapes accumulator feature in this tool (shown below the main calculator) lets you add several shapes and see the running total, making shape decomposition quick and accurate.

What is the difference between area and perimeter?

Area measures the two-dimensional space inside a shape — how much surface it covers — and is expressed in squared units (m², ft², etc.). Perimeter measures the total length of the shape's outer boundary and is expressed in linear units (m, ft, etc.). Think of it this way: if you were tiling a floor, you need area to know how many tiles to buy. If you were installing a skirting board along the walls, you need the perimeter to know how much board to purchase. This calculator displays both values simultaneously so you can use whichever measurement your project requires.

What is Heron's formula and when should I use it?

Heron's formula calculates a triangle's area using only the three side lengths, without needing to know the height. It is named after Hero of Alexandria (circa 60 AD). The formula is: A = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter. Use Heron's formula when you have measured or are given all three side lengths but cannot easily determine the perpendicular height — this is common in surveying, carpentry, and land measurement where you can string-measure sides but not a true vertical height. Select the SSS method in the Triangle Method selector to use it.

How accurate are the area calculations?

This calculator performs calculations in standard JavaScript floating-point arithmetic (IEEE 754 double precision), which provides approximately 15–17 significant decimal digits of precision. For practical measurement purposes this is far more than sufficient — even precision engineering rarely requires more than 6 significant figures. Results are displayed to 4–6 decimal places by default. The main source of error in real-world use is measurement inaccuracy, not computational rounding. A 1% error in a measured length causes approximately a 2% error in the calculated area, since area scales as the square of dimensions.

What is an annulus and how is its area calculated?

An annulus (also called a ring) is the region between two concentric circles — like a flat washer, a circular frame, or a cross-section of a pipe. It is defined by an outer radius R and an inner radius r. The area is calculated by subtracting the area of the inner circle from the area of the outer circle: A = π(R² − r²). This can also be written as A = π(R+r)(R−r). Common applications include calculating the area of circular ducts, pipes, ring-shaped gaskets, and annular flooring like round rugs with a hole. In this calculator, select Annulus (Ring) and enter both the outer and inner radius values.

How do I convert area between square feet and square meters?

Since 1 foot = 0.3048 meters, it follows that 1 square foot = 0.3048² = 0.09290304 square meters. To convert square feet to square meters, multiply by 0.0929. To convert square meters to square feet, multiply by 10.7639. This calculator's unit conversion table, displayed with every result, automatically shows the area in all common units simultaneously — mm², cm², m², km², in², ft², yd², and mi² — so you never need to convert manually. Just select your input unit, enter dimensions, and read off the value in whatever unit your project requires.