What unit does the area result use?

Whatever unit you put in, squared. Enter dimensions in metres → area in square metres. Inches → square inches. Feet → square feet. The calculator is unit-agnostic; it just multiplies the numbers you give it. Use a separate unit converter if you need to switch between systems.

Why does the rectangle area also show a perimeter?

Because most real-world area tasks need both. Flooring needs area only. Fencing needs perimeter only. Crown moulding and baseboard need perimeter. Wallpaper needs the room perimeter times the ceiling height. The calculator surfaces both for shapes that have a finite perimeter — rectangle, square, circle (as circumference) — so you don't run the math twice.

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

This calculator uses base × height. For three-sided input, you need Heron's formula: s = (a + b + c) / 2, then A = √(s × (s-a) × (s-b) × (s-c)). We will add a Heron's-formula triangle option in a future update. For now, drop a perpendicular from the apex to find the height.

What's the difference between a parallelogram and a rectangle here?

A rectangle is a special parallelogram where the angles are all 90°. The area formula is identical — base × height (or length × width). The difference matters for perimeter: a rectangle's perimeter is 2 × (length + width), while a parallelogram's perimeter depends on the slant side, which this calculator doesn't ask for. We omit parallelogram perimeter to avoid asking for a third input most users don't have.

How accurate is the π approximation in the circle and ellipse calculations?

We use JavaScript's Math.PI constant, which is accurate to 15 decimal places — more than enough for any practical area calculation. Results are rounded to 4 decimal places in the display, so a circle with radius 1 returns 3.1416 sq units (the true value is 3.14159265...).

Can the calculator handle very large dimensions (like acres or hectares)?

Yes. Enter the side lengths in feet → area in square feet (divide by 43,560 to convert to acres). Enter in metres → square metres (divide by 10,000 for hectares). For property planning, run the calc in feet or metres, then convert the output using a unit converter. Direct acre / hectare output is on the roadmap.

What is a circular sector — and when would I need its area?

A sector is a pie-slice-shaped piece of a circle, bounded by two radii and an arc. Area = (θ / 360) × π × r², where θ is the central angle in degrees. Common uses: pie-chart slice math, fan-shaped garden beds, sprinkler coverage area, and physics problems involving rotation. A 90° sector is exactly one-quarter of the full circle.

Does the area calculator handle composite or irregular shapes?

Not directly — but every composite shape can be decomposed into the eight primitives this calculator supports. For an L-shaped room: split it into two rectangles, run each, add the two areas. For a window with a semicircular top: rectangle plus half a circle. The 'sum of primitives' approach handles every shape geometry classifies as a polygon or a region bounded by circular arcs.

Where do these area formulas come from historically?

Egyptian rope-stretchers (≈2000 BCE) used the rectangle and triangle formulas for tax assessment on flooded Nile land. Euclid's Elements (≈300 BCE) systematized them in Book I. Archimedes calculated the area of a circle to high precision in the 3rd century BCE using the method of exhaustion — the precursor to integration. The formulas in this calculator are unchanged from then.

Why does the sector area max out at 360 degrees?

Because 360° is one full circle. A 'sector' larger than 360° is geometrically a circle plus a wrap — equivalent to (360° × full circles) + (remainder sector). Most practical uses cap the angle at 360°. The calculator returns an error above 360° to flag a likely input mistake; if you need wrap-around behaviour, calculate full circles separately.

How does this calculator's accuracy compare to manual / spreadsheet methods?

Identical to a spreadsheet using `=PI()` for circle / ellipse / sector, and exactly the same for rectangle / square / triangle / trapezoid / parallelogram, which involve no transcendental constants. The advantage of this calculator over spreadsheets is the shape-aware UI: you pick the shape once and the right inputs appear, instead of remembering which cells go where.

Is there a real-world example where I'd need ellipse area?

Oval running tracks (one straight section + two ellipse-end semicircles), elliptical swimming pools, decorative rugs, tabletops cut to an oval shape, and astronomical orbital math (orbits are ellipses, not circles). Most architectural drawings represent these as full ellipses for area calculation, then split into halves or quarters as needed.

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Free Area Calculator — Rectangle, Circle, Triangle, Trapezoid, Ellipse, Sector

Compute the area of eight common shapes from their dimensions — and get the perimeter or circumference where the shape supports it. Step-by-step working shown for every result.

Instant result
Private — nothing saved
Works on any device
AI insight included

Reviewed by CalcBold EditorialLast verified May 18, 2026Methodology

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What is Area?

Area is the measure of how much two-dimensional space a flat shape occupies. It is always reported in square units — square metres, square feet, square inches — because area combines two linear measurements (length × width, base × height, radius × radius) into a measurement of surface coverage. Whether you are buying tile, ordering paint, fencing a yard, or calculating the heat-loss surface of a building, every project that uses material to cover a flat region needs the area first.

The same shape can have different areas depending on how its perimeter encloses space. A 100-foot rope can enclose a 25-foot square (625 sq ft), a 1-foot × 49-foot rectangle (49 sq ft), or a circle of circumference 100 (≈796 sq ft). Among regular polygons with a fixed perimeter, the circle always encloses the most area — a fact called the isoperimetric inequality, and the reason cylinders and spheres dominate engineering whenever volume per surface matters.

The Eight Shape Formulas

Rectangle & square

A = length × width    (rectangle) A = side²              (square)

A square is a rectangle whose four sides are equal — same area formula, single dimension. Both also produce a perimeter: 2 × (length + width) for the rectangle and 4 × side for the square.

Source:NIST DLMF — Elementary Geometry· National Institute of Standards and Technology

Circle & circular sector

A = π × r²                    (full circle) A = (θ / 360) × π × r²        (sector of angle θ°)
circumference = 2 × π × r   (full circle)

The circle formula scales as r² — doubling the radius quadruples the area. The sector formula prorates that area by the angle: a 90° sector is exactly one-quarter of the full circle. The sector angle must be in degrees and between 0 and 360.

Source:Wolfram MathWorld — Circle· Wolfram Research

Triangle, trapezoid, parallelogram, ellipse

A = ½ × base × height        (triangle, base+height) A = (a + b) / 2 × h          (trapezoid) A = base × height            (parallelogram) A = π × a × b                (ellipse)

The triangle and parallelogram formulas share the base-times-height pattern; the triangle gets a one-half multiplier because two congruent triangles tile a parallelogram. The trapezoid formula averages the two parallel sides, then multiplies by the perpendicular height. The ellipse is the circle's generalization — when a = b = r, π × a × b collapses back to π × r².

Source:Wolfram MathWorld — Plane Geometry· Wolfram Research

Three Worked Examples

Concrete scenarios with specific numbers. Plug any of them into the calculator above to reproduce the step-by-step working.

Example 1

Carpeting a 15 × 12 ft living room

Shape: Rectangle
Length: 15 ft
Width: 12 ft

Apply the rectangle formula.
A = length × width = 15 × 12
Compute.
A = 180 sq ft
Add 10% waste factor (industry standard for carpet trim).
180 × 1.10 = 198 sq ft to order

Carpet needed: ~198 sq ft. Most carpet sells in 12-ft-wide rolls, so you'll likely buy a 12 × 17 ft piece (204 sq ft).

Carpet always wastes more than tile because the seam direction matters — order extra so the pattern runs unbroken.

Example 2

Pouring a circular patio, 14 ft diameter

Shape: Circle
Radius: 7 ft (diameter / 2)

Compute the radius from the diameter.
r = 14 / 2 = 7 ft
Apply the circle area formula.
A = π × 7² = π × 49
Multiply by π ≈ 3.14159.
A ≈ 153.94 sq ft
Convert to cubic yards of concrete at 4 inches deep.
153.94 × (4 / 36) / 27 ≈ 1.9 cu yd

Patio area: ~154 sq ft. Concrete needed at 4-inch depth: ~1.9 cubic yards (order 2 yd to cover waste).

Concrete is sold in 1/4-yard increments. Round up — running out mid-pour means a cold joint, which weakens the slab.

Example 3

Painting a trapezoidal gable wall

Shape: Trapezoid (gable end)
Bottom (wall width): 20 ft
Top (cut by roof): 8 ft
Height: 10 ft

Average the parallel sides.
(20 + 8) / 2 = 14 ft
Multiply by height.
A = 14 × 10 = 140 sq ft
Add the triangle peak above the trapezoid (if any) — none in this case.
Total = 140 sq ft
Paint coverage at 350 sq ft/gallon, 2 coats.
140 × 2 / 350 ≈ 0.8 gallons

Wall area: 140 sq ft. Paint needed: 1 gallon (rounded up — covers 2 coats with leftover for touch-ups).

Always buy at least 1 full gallon even for small jobs — opening a second can of a different lot risks colour mismatch.

How to Use This Calculator

Pick the shape from the dropdown. The dimension fields below will automatically relabel to ask for exactly what that shape needs.
Enter each dimension in whatever unit you have — centimetres, inches, feet, metres. The area output uses the same unit, squared. Use the unit converter if you need to switch.
Read the area at the top. For shapes that have a finite perimeter (rectangle, square, circle), that perimeter or circumference also appears below the area — saves a second run.
Tap Show working to see the formula substituted with your numbers, step by step. Useful for homework and double-checking.

Area Formulas at a Glance

The eight shapes this calculator supports, paired with the formula it uses and the perimeter (or circumference) it also reports where applicable.

Shape reference

Area + perimeter formulas for eight common shapes

Area + perimeter formulas for eight common shapes
Scenario	Area formula	Perimeter / extra	Typical use
Rectangle	A = length × width	2 × (length + width)	Rooms · walls · sheets
Square	A = side²	4 × side	Tiles · plots · grids
Circle	A = π × r²	Circumference = 2 × π × r	Patios · pools · cake tins
Triangle (base × height)	A = ½ × base × height	(varies — needs all 3 sides)	Roof gables · sails
Trapezoid	A = (a + b) / 2 × h	(needs both slant sides)	Gable walls · land plots
Parallelogram	A = base × height	(needs the slant side)	Banners · diamond plots
Ellipse	A = π × a × b	Ramanujan approx for perimeter	Oval tracks · pools · tabletops
Circular sector	A = (θ / 360) × π × r²	Arc length = (θ / 360) × 2π × r	Pie slices · sprinkler zones

Perimeter is shown for shapes where it can be computed from the same dimensions the area uses. Triangle and parallelogram perimeters need the slant sides, which this calculator does not currently ask for.

Common Mistakes When Calculating Area

Using the diameter as the radius. Circle area is π × r², not π × d². If you have the diameter (e.g. a 14-foot patio), divide by 2 first. This single mistake quadruples the result.
Mixing units mid-calculation.Don’t put length in feet and width in inches. Convert to the same unit first. The calculator does not unit-check inputs — wrong units in, wrong units out.
Skipping the waste factor on real-world orders. Tile waste 10%, carpet waste 10–15%, paint over-purchase 10%, drywall waste 10%. The bare area is the theoretical minimum; always order more than the calculator says.
Forgetting to subtract openings. Painting a wall? Subtract windows and doors. Tiling a floor? Subtract built-in cabinets. The calculator gives you the gross area; net area for the order is the gross minus the openings.
Using base × height for a triangle’s slant length instead of perpendicular height.The “height” in the triangle formula is the perpendicular distance from the base to the apex — not the slant length. For a right triangle this is the same; for an obtuse triangle it is not.
Treating composite shapes as their largest enclosing rectangle. An L-shaped room is not a rectangle. Decompose it into two rectangles, compute each, add the areas. The calculator handles each piece; you combine them.
Confusing sector area with arc length.A sector has both an area (region) and an arc length (1D distance along the curved edge). Don’t use the sector area when the project needs the arc length (e.g. fencing a curved garden bed).

Area Terminology — Quick Reference

Eight terms that appear across geometry problems and real-world measurement tasks. Skim the snippet line; expand the card for the longer explanation.

Quick reference

Area glossary

Area Two-dimensional surface coverage of a shape. Always reported in square units (m², ft², in²).: Computed as the product of two linear dimensions (length × width) or as a function of one dimension and a constant (π × r² for a circle).
Perimeter The total length around the outside of a shape. Reported in linear units (m, ft, in).: Used for fencing, trim, edging, and any task that follows the boundary. Different formulas per shape: 2(l+w) for rectangle, 2πr for circle (called circumference), sum of all sides for polygons.
Radius The distance from the centre of a circle (or sphere) to its edge. Equals half the diameter.: All circle and ellipse formulas use radius, not diameter. If a problem gives you the diameter (most everyday measurements), divide by 2 before applying the formula.
Diameter The straight-line distance across a circle through its centre. Equals 2 × radius.: Most consumer products quote diameter (pizza is 12 inches diameter, not 6 inches radius). Always convert to radius before plugging into the area formula.
π (pi) The mathematical constant ≈ 3.14159. The ratio of a circle's circumference to its diameter.: Irrational and transcendental — its decimal expansion never repeats or terminates. JavaScript's Math.PI is accurate to 15 decimal places, which suffices for any real-world area calculation.
Source: NIST — Fundamental constants
Heron's Formula Computes triangle area from the three side lengths alone — no height needed. A = √(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2.: Named for Hero of Alexandria (1st century CE). Useful when you can measure a triangle's sides but not its height — surveying, sailmaking, irregular land plots. This calculator does not yet support Heron-style input; use base × height for now.
Isoperimetric Inequality Among all shapes with a fixed perimeter, the circle encloses the most area. The deep reason cylinders and spheres dominate engineering.: Mathematically: for any plane figure with perimeter L, area ≤ L² / (4π), with equality only for the circle. The same principle in 3D explains why soap bubbles are spherical and why high-pressure tanks are cylindrical.
Square Units The unit of area. Always the input unit, squared (sq m, sq ft, sq in).: If you input centimetres, the area comes back in square centimetres. To convert: 1 sq ft = 0.0929 sq m, 1 acre = 43,560 sq ft, 1 hectare = 10,000 sq m. The Britannica Plane-Geometry article has the full table.
Source: Britannica — Plane Geometry

When to Use Area vs Perimeter vs Volume

Real projects rarely need just one. Match the measurement to the task:

Area — for material that covers a 2D surface: paint, wallpaper, carpet, tile, sod, drywall, fabric, sheet metal.
Perimeter — for material that follows a 1D boundary: fence, trim, baseboard, crown moulding, edging, sealant bead, gutter.
Volume — for material that fills 3D space: concrete, mulch, gravel, paint thickness × area, water for pools or tanks. Use the dedicated unit converter + area × depth, or the shape-specific calcs: concrete, paint, drywall, fence.

Background

A Brief History of Area Measurement

Area measurement is among the oldest applied mathematics. Egyptian 'rope-stretchers' (harpedonaptae) used standardized lengths of knotted rope to measure rectangular plots along the Nile flood plain as early as 2000 BCE, primarily for tax assessment after the annual flood erased boundary markers ^[1]. The Rhind Mathematical Papyrus (≈1650 BCE) contains explicit area formulas for triangles, rectangles, and circles, plus a remarkably accurate approximation for π as 256/81 ≈ 3.16.

Euclid's Elements (≈300 BCE) systematized plane geometry in Book I, deriving the rectangle and parallelogram area formulas as consequences of the parallel postulate. Archimedes (3rd century BCE) calculated the area of a circle to high precision using the method of exhaustion — inscribing and circumscribing polygons with increasing numbers of sides, bounding π between 3 + 10/71 and 3 + 1/7 ^[2]. His method is the geometric precursor to integral calculus, which Newton and Leibniz formalized 1900 years later.

Modern area measurement standards come from the metric system established under Napoleon (1799) and codified internationally by the Bureau International des Poids et Mesures (BIPM) ^[3]. The square metre is the SI derived unit; everything else (acres, hectares, square feet) is defined relative to it. In the United States, the survey foot persisted alongside the international foot until 2022, when NIST officially retired the survey foot in favour of a single unified foot definition — affecting century-old property descriptions and surveying records.

MacTutor — Egyptian Mathematics · MacTutor History of Mathematics, University of St Andrews
Archimedes — Measurement of a Circle · MacTutor History of Mathematics, University of St Andrews · 3rd c. BCE
BIPM SI Brochure — The International System of Units · Bureau International des Poids et Mesures

Related Math Tools

Area is one corner of measurement geometry. Pair this calculator with the unit converter when switching between metric and imperial, the percentage calculator when adding waste factors, and the ratio calculator when scaling drawings. For material-volume tasks downstream of area (concrete, paint, mulch), use the dedicated Home Improvement calculators — they bake the area × depth × waste factor logic into one step.

Sources & Methodology

The formulas, thresholds, and benchmarks behind this calculator are anchored to the primary sources below. Where a study or agency document is the underlying authority, we link straight to it — not a summary or republished version.

NIST DLMF — Elementary Geometry· National Institute of Standards and Technology
Authoritative reference for the elementary functions (π, trig, square root) that underlie every area formula in the calculator.
Accessed 2026-05-18
Wolfram MathWorld — Plane Geometry· Wolfram Research
Standard reference for the algebraic identities producing area, perimeter, and circumference of rectangles, circles, triangles, trapezoids, parallelograms, ellipses, and sectors.
Accessed 2026-05-18
BIPM SI Brochure — Units of Area· Bureau International des Poids et Mesures
Defines square metre as the SI derived unit of area and the conventions for compound units (square foot, hectare, acre) used in the calculator's unit-agnostic output.
Accessed 2026-05-18
Britannica — Geometry: Areas of Plane Figures· Encyclopaedia Britannica
Encyclopaedic overview of the historical development of area formulas — Egyptian rope-stretchers, Euclid's Elements Book I, Archimedes' calculation of circular area.
Accessed 2026-05-18

Frequently Asked Questions

The most common questions we get about this calculator — each answer is kept under 60 words so you can scan.

What unit does the area result use?
Whatever unit you put in, squared. Enter dimensions in metres → area in square metres. Inches → square inches. Feet → square feet. The calculator is unit-agnostic; it just multiplies the numbers you give it. Use a separate unit converter if you need to switch between systems.
Why does the rectangle area also show a perimeter?
Because most real-world area tasks need both. Flooring needs area only. Fencing needs perimeter only. Crown moulding and baseboard need perimeter. Wallpaper needs the room perimeter times the ceiling height. The calculator surfaces both for shapes that have a finite perimeter — rectangle, square, circle (as circumference) — so you don't run the math twice.
How do I find the area of a triangle when I only know the three sides?
This calculator uses base × height. For three-sided input, you need Heron's formula: s = (a + b + c) / 2, then A = √(s × (s-a) × (s-b) × (s-c)). We will add a Heron's-formula triangle option in a future update. For now, drop a perpendicular from the apex to find the height.
What's the difference between a parallelogram and a rectangle here?
A rectangle is a special parallelogram where the angles are all 90°. The area formula is identical — base × height (or length × width). The difference matters for perimeter: a rectangle's perimeter is 2 × (length + width), while a parallelogram's perimeter depends on the slant side, which this calculator doesn't ask for. We omit parallelogram perimeter to avoid asking for a third input most users don't have.
How accurate is the π approximation in the circle and ellipse calculations?
We use JavaScript's Math.PI constant, which is accurate to 15 decimal places — more than enough for any practical area calculation. Results are rounded to 4 decimal places in the display, so a circle with radius 1 returns 3.1416 sq units (the true value is 3.14159265...).
Can the calculator handle very large dimensions (like acres or hectares)?
Yes. Enter the side lengths in feet → area in square feet (divide by 43,560 to convert to acres). Enter in metres → square metres (divide by 10,000 for hectares). For property planning, run the calc in feet or metres, then convert the output using a unit converter. Direct acre / hectare output is on the roadmap.
What is a circular sector — and when would I need its area?
A sector is a pie-slice-shaped piece of a circle, bounded by two radii and an arc. Area = (θ / 360) × π × r², where θ is the central angle in degrees. Common uses: pie-chart slice math, fan-shaped garden beds, sprinkler coverage area, and physics problems involving rotation. A 90° sector is exactly one-quarter of the full circle.
Does the area calculator handle composite or irregular shapes?
Not directly — but every composite shape can be decomposed into the eight primitives this calculator supports. For an L-shaped room: split it into two rectangles, run each, add the two areas. For a window with a semicircular top: rectangle plus half a circle. The 'sum of primitives' approach handles every shape geometry classifies as a polygon or a region bounded by circular arcs.
Where do these area formulas come from historically?
Egyptian rope-stretchers (≈2000 BCE) used the rectangle and triangle formulas for tax assessment on flooded Nile land. Euclid's Elements (≈300 BCE) systematized them in Book I. Archimedes calculated the area of a circle to high precision in the 3rd century BCE using the method of exhaustion — the precursor to integration. The formulas in this calculator are unchanged from then.
Why does the sector area max out at 360 degrees?
Because 360° is one full circle. A 'sector' larger than 360° is geometrically a circle plus a wrap — equivalent to (360° × full circles) + (remainder sector). Most practical uses cap the angle at 360°. The calculator returns an error above 360° to flag a likely input mistake; if you need wrap-around behaviour, calculate full circles separately.
How does this calculator's accuracy compare to manual / spreadsheet methods?
Identical to a spreadsheet using `=PI()` for circle / ellipse / sector, and exactly the same for rectangle / square / triangle / trapezoid / parallelogram, which involve no transcendental constants. The advantage of this calculator over spreadsheets is the shape-aware UI: you pick the shape once and the right inputs appear, instead of remembering which cells go where.
Is there a real-world example where I'd need ellipse area?
Oval running tracks (one straight section + two ellipse-end semicircles), elliptical swimming pools, decorative rugs, tabletops cut to an oval shape, and astronomical orbital math (orbits are ellipses, not circles). Most architectural drawings represent these as full ellipses for area calculation, then split into halves or quarters as needed.