Area Calculator — 20 Shapes with a Draggable Live Diagram, Composite Builder & Polygon Drawing

What is area?

Area is the amount of flat, two-dimensional surface a shape covers. It is measured in square units — square centimetres (cm²), square metres (m²), square inches (in²), square feet (ft²), square yards (yd²), acres, or hectares. A rectangle 5 m long and 3 m wide covers 15 m² of floor. A circle of radius 5 cm covers about 78.54 cm² of paper.

Area is different from perimeter. Perimeter measures the distance around the edge of a shape (a length: cm, m, ft). Area measures the surface enclosed (a squared length: cm², m², ft²). A 3×4 m rectangle has a perimeter of 14 m and an area of 12 m². You buy paint and flooring by area; you buy fencing and skirting by perimeter.

Area is also different from volume, which measures three-dimensional space (cm³, m³). A 1 m × 1 m × 1 m cube has a surface area of 6 m² and a volume of 1 m³. When you paint the outside of a box you need the surface area; when you fill the inside you need the volume.

How to use this calculator

1. Pick a mode. Shapes gives you 20 regular shapes with their standard formulas. Composite lets you add and subtract pieces for L-shaped rooms and cut-outs. Draw Polygon lets you click points on a canvas and computes the area with the shoelace formula.
2. Pick a shape. Use the popular-shape row for the six most common shapes, or filter by category — Basic, Triangles, Circles, Quadrilaterals, Polygons. The diagram and inputs update immediately.
3. Drag, type, or both. Every shape on the diagram has amber handles. Drag any handle with your mouse or finger to reshape the figure — the inputs, area, perimeter and every output update live. Release to finish. Or type exact numbers in the input fields; the diagram redraws as you type.
4. Choose units. Type in mm, cm, m, km, in, ft, yd or mi. Switch input format to Feet & inches to type tape-measure notation like 12' 6" or 12 ft 6 in. The to-scale diagram shows the real proportions.
5. Zoom and pan. Scroll over the diagram to zoom between 25% and 500%. Drag empty space to pan. Use the + / − / percentage buttons in the corner. Double-click the canvas to reset.
6. Toggle overlays. Grid, labels, diagonals and radii can each be turned on or off independently, so you can get a clean screenshot or a fully annotated teaching diagram.
7. Estimate cost. Open the cost estimator to multiply the area by a price per square unit and add a waste allowance. This is the standard budgeting step for paint, tiles, flooring or land.
8. Export. Copy the results as text, share a link that reproduces your exact inputs, or download the diagram as PNG or SVG.

The main area formulas

Drag-to-resize — how the interactive diagram works

Every shape in this calculator has amber handles on its corners or edges. Each handle is a "grab point" for a specific measurement — drag the handle with your mouse (or finger on a mobile device) and the corresponding input value changes, which cascades into the diagram, the area, the perimeter and every other output.

Here is what each handle controls on each shape:

Drag any handle and watch the inputs update live. Release the handle when you have the size you want — the drag has continuous values, so you can hit exact targets by nudging with your arrow keys in the input fields afterwards. This is the fastest way to find the size that looks right without doing any arithmetic.

Squares and rectangles

A rectangle has four right angles and two pairs of parallel sides. Its area is the product of its length and width:

A rectangle 5 m by 3 m covers 15 m². If you measured the sides in feet, the answer is in square feet. The square is a rectangle where all four sides are equal, so A = a². A square with 4 m sides has an area of 16 m².

The main trap with rectangles is mixing units. A length of 3 m and a width of 200 cm does not give 600 m² — you must convert one to match the other first. This calculator's unit selector applies to every field at once, so you cannot accidentally mix them.

Real-world rectangles: the floor of a rectangular room, a sheet of plywood, a football pitch, a phone screen, a window pane. Every one of these is solved with the same multiplication.

Triangles

A triangle has three sides and three angles. Its area depends on which measurements you have, and there are four common formulas:

Base and height

A = ½ × base × height. The height must be perpendicular to the base, not a slanted side. For a triangle with base 8 cm and height 5 cm, A = 20 cm².

Three sides (Heron's formula)

If you know all three sides a, b and c, first compute the semiperimeter s = (a + b + c) / 2, then:

For sides 3, 4 and 5 cm: s = 6, and A = √(6 × 3 × 2 × 1) = √36 = 6 cm². Heron's formula is exact for any triangle, including obtuse ones, as long as the three sides can actually form a triangle (the sum of any two sides must exceed the third).

Two sides and the included angle (SAS)

A = ½ × a × b × sin(C), where C is the angle between sides a and b. For a = 5 cm, b = 7 cm, C = 40°: A = ½ × 5 × 7 × 0.6428 ≈ 11.25 cm².

Equilateral triangle

All three sides are equal, so A = (√3 / 4) × a². A 6 cm equilateral triangle has an area of 15.59 cm².

Circles, sectors and segments

A circle is the set of points at a fixed distance (the radius r) from a centre point. Its area is:

For r = 5 cm: A = π × 25 ≈ 78.54 cm². If you know the diameter d instead, the radius is d/2, so the formula becomes A = π × (d/2)².

Semicircle and quarter circle

A semicircle is exactly half a circle: A = ½ × π × r². A quarter circle is exactly a quarter: A = ¼ × π × r². Both appear constantly in architecture and engineering — arched windows, pipe cross-sections, corner fillets.

Sector

A sector is a "pie slice" bounded by two radii and an arc. If the central angle is θ degrees:

For r = 6 cm and θ = 60°: A = (60/360) × π × 36 ≈ 18.85 cm². A full circle is θ = 360°, which gives back πr² — a useful sanity check.

Segment

A segment is the region between a chord and the arc it cuts off. Its area is the sector area minus the triangle formed by the two radii and the chord:

For a segment of a 5 cm circle cut off by a 60° chord: θ = π/3 ≈ 1.047 rad, sin θ ≈ 0.866, A = ½ × 25 × (1.047 − 0.866) ≈ 2.26 cm². Segment areas are much smaller than the sector for the same angle because the triangle takes up most of the pie slice.

Ellipse and annulus

An ellipse is a stretched circle with two semi-axes a (horizontal) and b (vertical). Its area is:

For a = 5 cm and b = 3 cm: A = π × 15 ≈ 47.12 cm². An ellipse with a = b is just a circle, and the formula reduces to πa² — another sanity check.

An annulus (ring) is a circle with a smaller circle removed from the centre. Its area is the difference of the two circle areas:

For R = 5 cm and r = 3 cm: A = π × (25 − 9) = 50.27 cm². Annuli appear in washers, O-rings, pipes, donuts, and the coloured rings of a target.

Parallelograms, trapezoids, rhombuses and kites

Parallelogram

A parallelogram has two pairs of parallel sides. Its area is A = base × perpendicular height, not base × slanted side. For a parallelogram with base 8 cm and height 5 cm, A = 40 cm². The slant length is used for the perimeter, not the area.

Trapezoid

A trapezoid has exactly one pair of parallel sides. Its area is the average of the parallel sides times the height:

For a = 6, b = 10 and h = 4: A = ½ × 16 × 4 = 32. The formula works for any trapezoid, including right trapezoids (one leg perpendicular to the bases) and isosceles trapezoids (equal legs).

Rhombus

A rhombus is a parallelogram with all four sides equal. Its area is half the product of its diagonals:

For diagonals 8 cm and 6 cm: A = 24 cm². If you know the side length and interior angle instead, A = a² × sin(θ).

Kite

A kite has two pairs of adjacent equal sides. Its area uses the same formula as the rhombus:

The diagonals of a kite are perpendicular, which is why the formula simplifies to half their product. For diagonals 10 cm and 6 cm: A = 30 cm².

Regular polygons

A regular polygon has all sides equal and all interior angles equal. Its area is half the perimeter times the apothem (the perpendicular distance from the centre to a side):

Equivalently, for an n-gon of side s:

The apothem is a = s / (2 × tan(π/n)), the circumradius is R = s / (2 × sin(π/n)), and the interior angle is (n − 2) × 180° / n. The calculator shows all of these in the property panel.

Common regular polygons: hexagon (n = 6, honeycomb cells, many tiles), octagon (n = 8, stop signs), pentagon (n = 5, the US Department of Defense building). For n = 3 the formula reduces to the equilateral triangle; for n = 4 it gives the square.

Composite and irregular shapes

Most real-world surfaces are not a single regular shape. A room might be L-shaped. A garden might be a rectangle with a semicircular patio attached. A plot of land might be an irregular quadrilateral. There are two general approaches:

Decomposition (add and subtract)

Break the shape into regular pieces. Compute each piece's area. Add pieces that are present; subtract pieces that are holes or cut-outs. An L-shaped room can be split into two rectangles. A rectangular wall with a window is the wall area minus the window area. This is exactly what the Composite mode does: add positive pieces and negative pieces, and read the net total.

The shoelace formula (coordinates)

If you know the (x, y) coordinates of the corners in order around the shape, the shoelace (or Gauss) formula gives the exact area of any simple polygon, regular or irregular:

Take the vertices in order around the shape and wrap the last point back to the first. The absolute value handles clockwise vs anticlockwise ordering. The Draw Polygon mode implements the shoelace formula on the points you click on the canvas, so you can trace any shape and get its exact area.

For very irregular shapes — a lake on a map, a stain on a floor, a property with curved boundaries — neither method is exact unless you approximate the curves with many small straight segments. The Draw Polygon mode lets you place as many vertices as you need, so a curved boundary can be approximated as closely as you like by clicking more points along it.

Area units and conversions

Because area is a squared length, every length conversion factor is squared when you convert an area. 1 metre is 3.28084 feet, so 1 square metre is 3.28084² = 10.7639 square feet. Getting this wrong — converting area by the length factor rather than its square — is the single most common unit error in construction and real estate.

⚠ Squared conversions. 1 square yard is 9 square feet (not 3). 1 square metre is about 10.76 square feet (not 3.28). Whenever you convert an area, square the length conversion factor. This is why a "200 square foot" room is much smaller than "200 feet by 200 feet".

Feet, inches, and tape-measure input

In the US, UK, Canada, Australia and much of the Commonwealth, construction measurements are still commonly taken in feet and inches. A tape measure reads 12 feet 6 inches, not 12.5 feet. Writing that as a decimal requires converting the inches: 6 inches ÷ 12 = 0.5 feet, so the length is 12.5 feet.

This calculator's Feet & inches input mode parses the tape-measure notation directly. You can type:

• 12' 6" — the standard notation with foot and inch marks
• 12 ft 6 in — spelled out
• 12' 6 — inches without the inch mark
• 12.5 — decimal feet (still works)

The calculator converts everything to decimal feet internally, computes the area, and shows the result in square feet alongside square metres, square inches and square yards. For a room 12' 6" by 10' 3", the area is 128.125 ft² ≈ 11.90 m².

Worked examples

Example 1 — a rectangular room

A living room is 5.2 m long and 4.0 m wide. A = 5.2 × 4.0 = 20.8 m². If you are buying flooring at £18 per m², the material cost is £374.40 before waste. With 10% waste for cuts, the order is 22.88 m², costing £411.84.

Example 2 — a circular table

A round table has a diameter of 1.2 m, so the radius is 0.6 m. A = π × 0.6² ≈ 1.13 m². A tablecloth needs to be at least this size — 1.2 m² is a reasonable order to allow for a little overhang.

Example 3 — a triangular sail

A sail has a base of 3.5 m and a height of 4.2 m. A = ½ × 3.5 × 4.2 = 7.35 m². Sailcloth is sold by the square metre, so you need 7.35 m² plus seams.

Example 4 — an L-shaped room

An L-shaped room is made of two rectangles: 4 m × 3 m and 2 m × 2.5 m. Areas: 12 m² and 5 m². Total = 17 m². If a circular column of radius 0.3 m stands in the room, subtract π × 0.3² ≈ 0.28 m², giving 16.72 m² of usable floor. The Composite mode does this in one pass.

Example 5 — a plot of land

A plot is a quadrilateral with corners at (0, 0), (30, 0), (25, 20) and (5, 18) metres. The shoelace formula gives 540 m². Converting: 540 m² = 5,812.5 ft² = 0.1335 acres = 0.054 hectares. Draw Polygon mode computes this exactly from the coordinates you click.

Common mistakes

• Using a slanted side as the height. For triangles, parallelograms and trapezoids, the height must be perpendicular to the base. Using the slant length inflates the area by 10–40% depending on the angle.
• Forgetting to square the unit conversion. 1 m² is 10.76 ft², not 3.28 ft². Square the length factor.
• Mixing units in one calculation. A length in metres and a width in centimetres gives a nonsensical result. Convert to one unit first — this calculator applies the unit selector to every field.
• Confusing area and perimeter. Paint covers area (m²); skirting board runs along the perimeter (m). Buying 20 m of skirting is not the same as buying 20 m² of paint.
• Forgetting the cut-outs. A wall with a window has less paint area than the full wall. Subtract window and door areas before ordering materials.
• Using the diameter where the radius is needed. Circle area uses r, not d. If you have the diameter, halve it first.
• Using degrees in a radians formula (or vice versa). Sector area uses θ/360 when θ is in degrees, but ½r²θ when θ is in radians. The two are not interchangeable.
• Assuming a plot is rectangular. Many real boundaries are not parallel. If the shape is irregular, use the shoelace formula or Draw Polygon mode rather than treating it as a rectangle.
• Ignoring waste. Tiles, flooring and paint always need 5–15% extra for cuts, mistakes and future touch-ups. The cost estimator's waste field exists for exactly this reason.

Where area calculations show up

• Flooring and tiling. Order tiles or boards by the room's area plus 10% waste. A 20 m² room needs about 22 m² of material.
• Painting and decorating. Paint coverage is quoted in m² per litre. A wall of 30 m² needs about 4 litres of a typical emulsion (two coats, 15 m²/L).
• Roofing. Roof area is not the same as floor area — a pitched roof has more surface. The area of each roof plane is a trapezoid or triangle, and the sum is what you buy in shingles or tiles.
• Land and property. Plot sizes are quoted in square metres, square feet, acres or hectares. Surveyors use the shoelace formula on the boundary coordinates, which is exactly what Draw Polygon mode does.
• Agriculture. Seed, fertiliser and pesticide application rates are per hectare or per acre. Field area determines the quantity of everything you apply.
• Engineering and manufacturing. Sheet metal, fabric, glass and plastic are ordered by area. Cutting a batch of parts from a sheet is a nesting problem whose objective is to minimise wasted area.
• Physics and chemistry. Cross-sectional area appears in pressure (P = F/A), drag (F = ½ρv²C_dA), and heat transfer (Q = hAΔT).
• Cartography and GIS. Every map polygon has an area attribute, computed by the shoelace formula or a geodesic equivalent.

Area in real estate and land

Property listings in the US quote square footage, in the UK quote square metres, in Australia quote "squares" (one square = 100 ft² = 9.29 m²), in India quote "square feet" or "square yards" depending on region, and in continental Europe quote square metres. Converting between them without the squared unit conversion is one of the most common causes of pricing errors.

A practical worked example: a US listing says "1,850 square feet". In the UK that is 1,850 ÷ 10.7639 = 171.9 m². If the price is $380 per square foot, it is $4,090 per m² — you would need to multiply by 10.7639 again to compare with a €4,200/m² Paris listing.

Land areas use larger units because they are measuring outdoor spaces. 1 acre = 4,840 square yards = 43,560 square feet = 4,046.86 m² = 0.4047 hectares. A 100 m × 200 m plot is 20,000 m², which is 4.94 acres or 2 hectares. The calculator shows all four units simultaneously, so you never have to make the conversion by hand.

Odd-shaped plots are increasingly common as subdivisions and boundary relocations happen. For these, the Draw Polygon mode is faster than any arithmetic approach: trace the boundary on a scaled grid and read the area from the shoelace result. For a boundary with a curved road frontage, add three or four extra vertices along the curve — the polygon area converges to the true area as the vertex count increases.

Area for painting and wallpaper

Painting a room looks like a rectangle problem but is rarely that simple. The gross wall area is (2 × (length + width) × height), then subtract doors and windows:

For a 5 m × 4 m room with 2.7 m ceilings, one 2 m × 0.8 m door and two 1.2 m × 1 m windows:

• Gross wall area: 2 × (5 + 4) × 2.7 = 48.6 m²
• Door area: 2 × 0.8 = 1.6 m²
• Windows: 2 × 1.2 × 1.0 = 2.4 m²
• Net paintable: 48.6 − 1.6 − 2.4 = 44.6 m²

Two coats of emulsion covering 14 m²/L requires 44.6 × 2 ÷ 14 = 6.4 L, so buy two 4 L tins (with overlap) or one 5 L plus one 2 L. Add 5–10% for cutting in around edges, and round up to the nearest tin size.

Wallpaper is quoted in rolls, not m². A standard UK roll covers 5.2 m²; a US double roll covers 5.6 m². Divide the net wall area by the roll coverage and add 10–15% for pattern matching. Anaglypta and other embossed papers usually need their own coverage figure.

Area for flooring and tiling

Flooring and tiling have their own waste rules. Straight-lay tile typically wastes 5–10%; diagonal-lay wastes 10–15%; herringbone can waste up to 20% because of the many angled cuts. Always add waste before rounding up to the nearest box — a box that is 1.44 m² cannot be broken in most stores, so 15.5 m² of tiles should become 12 boxes of 1.44 m² = 17.28 m², not 11 boxes.

For a rectangular room, measure both walls at floor level (rooms are not always square) and multiply. For an L-shaped room, split into two rectangles and add. For a room with a bay window, add the area of the bay — usually a trapezoid or semicircle — as a separate shape in Composite mode. For a curved wall, use the segment formula.

A standard 60 × 60 cm tile is 0.36 m². A 12 m² room needs 12 ÷ 0.36 = 33.3 tiles, so 34 tiles for a straight lay, or 37–38 for a diagonal lay. The cost estimator on this page multiplies tile price × net area × (1 + waste%) to give a total cost before adhesive and grout.

A short note on the history of area

Area measurement is one of the oldest branches of applied mathematics. Egyptian surveyors used area formulas to re-establish field boundaries after the Nile floods, and the Rhind Mathematical Papyrus (c. 1650 BCE) contains correct formulas for triangles, rectangles and circles (with π approximated as 256/81 ≈ 3.16).

The Greek mathematicians turned area calculation into a deductive science. Eudoxus of Cnidus (c. 370 BCE) developed the method of exhaustion to prove the area of a circle, and Archimedes (c. 250 BCE) used it to prove that the area of a circle equals πr² and to compute the area of a parabolic segment. Archimedes' Quadrature of the Parabola is effectively the first rigorous integration.

The shoelace formula for polygon area has been known since at least the 18th century and is standard in surveying and CAD. It works for any simple polygon — convex, concave, or with hundreds of vertices — as long as the vertices are listed in order around the boundary.

Privacy

Every calculation, drawing, conversion, export and share-link runs in your browser with plain JavaScript. Nothing you type is uploaded, logged or sent to a server. There is no account and no network request after the page loads. Recent calculations are optionally kept in your browser's localStorage so you can reload them, and clearing them removes them permanently from your device.

Limitations

• Every shape in the calculator is treated as a flat 2D surface. It does not compute surface area of 3D objects — use the volume calculator for that.
• Curved shapes (circle, ellipse, sector, segment) use exact formulas for area but numerical approximations for some perimeters. The ellipse perimeter uses Ramanujan's approximation, accurate to better than 0.01% for most axis ratios.
• The composite mode sums the areas of pieces in any layout, but it does not check whether the pieces overlap visually. It is a numerical sum, not a spatial union.
• The Draw Polygon mode computes area from the clicked points; it does not detect or prevent self-intersecting polygons. If you cross the lines, the shoelace formula gives the signed area of the enclosed regions, which may not match your intention.
• Feet-and-inches input is parsed but not validated against the nearest 1/16 inch — you can type 12' 6.3" and it will use the decimal value.
• Extremely large or small numbers lose precision in the last digits; results are formatted to a sensible number of significant figures.

Frequently asked questions

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