Area Calculator
Fourteen flat shapes, with the perimeter and the land units beside every answer
Same edge, very different area
| Shape | Area enclosed | Share of the best |
|---|
This is the question behind most fencing and edging jobs and almost nobody answers it with numbers. Take a fixed length of edge and the shape you bend it into decides how much ground you get. A circle always wins, a square is the best rectangle, and a long thin strip is a disaster: 40 metres of fence gives 127 square metres as a circle and only 36 as an 18 by 2 rectangle.
Same area, very different edge
| Layout | Edge needed | What it looks like |
|---|
Run the same comparison the other way and it explains why land is priced by area but fenced by the metre. One acre is one acre whatever shape it takes, yet a 40 to 1 ribbon of it needs more than three times the fencing of a square.
Area is flat, volume is not. Everything here multiplies two lengths and lands in square units. The moment a third length joins in you are measuring space rather than ground, and that belongs on the volume calculator instead.
Which shape is the thing you are measuring?
Every area calculator on the first page of results opens a dropdown and assumes you already know the answer. Most of the time you do not, and picking the wrong shape costs far more than any rounding. This table maps ordinary spaces onto the shape that actually describes them, and says which tape measurements to take.
| What you are measuring | Use this shape | What to measure, and the trap |
|---|---|---|
| A bedroom, a patio, a sheet of plywood | Rectangle | Length and width along the skirting, not across the diagonal. Measure at floor level, because walls lean. |
| A lounge with a chimney breast or a bay | L-shape | Overall length and width first, then the cutout. A chimney breast is a cutout you subtract, a bay is a second rectangle you add. |
| A gable end wall, the triangle above the eaves | Triangle, base and height | The width of the house and the vertical rise to the ridge. The rafter length is the slope, not the height, and using it inflates the answer. |
| A triangular corner plot you walked around | Triangle, three sides | All three boundary lengths. You never need to find a height, which is why this mode exists. |
| A cathedral wall, one side taller than the other | Trapezoid | The width, then both heights. This is the shape a sloping ceiling gives you, and the two heights go in as the two bases with the width as the height. |
| A drive that widens as it leaves the road | Trapezoid | Width at the gate, width at the house, and the distance between them measured straight, not along a curving kerb. |
| A round lawn, a circular pond, a fire pit | Circle | The radius, which is half the distance across. Measuring the full width and forgetting to halve it makes the answer four times too big. |
| A path or border running round a pond | Ring | Outer radius and inner radius. Subtracting circles is far easier than trying to treat the path as a long bent rectangle. |
| A quarter circle flower bed in a corner | Sector | Radius and the angle at the corner. A right angled corner is 90 degrees, so a quarter of the circle. |
| An oval rug, table or planting bed | Ellipse | Half the long width and half the short width. Both are semi axes, so halve each one before entering it. |
| A leaning panel, a skewed parking bay | Parallelogram | Base and the height at right angles to it. The slanted side is longer than the height and using it overstates the area. |
| A hexagonal gazebo base or paving slab | Regular polygon | The number of sides and the length of one of them. Works for anything from a triangle to a near circle. |
| A diamond shaped inset, a kite of glass | Rhombus or kite | The two diagonals, corner to opposite corner. Both of these are half the product of the diagonals. |
| A field with a curved or ragged boundary | Split it up | No single shape fits. Cut it into rectangles and triangles, work out each, then add. The method is below. |
The grey row is not a picker option because it is a method rather than a formula. It is also the most useful row in the table, so it gets its own section.
Splitting an irregular plot into shapes that behave
Real ground is rarely one clean shape. The professional answer for a surveyed plot is the shoelace formula, which turns a list of corner coordinates into an area, and it is what land surveyors have used for two centuries. It needs coordinates, though, and nobody standing in a field with a tape measure has coordinates. What they have is side lengths, which means the practical method is decomposition: cut the plot into pieces you can measure, work out each piece, then add and subtract.
- 1Sketch it and draw in the cuts
Rough is fine, proportion is not. Slice the plot with straight lines into rectangles and triangles, aiming for as few pieces as possible and for cuts that run between points you can actually reach with a tape.
- 2Number every piece and measure it fully
Write the measurements on the sketch as you take them, not afterwards. A rectangle needs two, a triangle needs three sides or a base and a height at right angles to it.
- 3Work each piece in the same unit
Run every piece through the calculator above without changing the unit between pieces. Mixing feet and inches halfway through is the single most common failure in this whole method.
- 4Add the pieces, subtract the holes
A pond, a patio you are not turfing, a chimney breast and a doorway are all holes. Work them out the same way and take them off the total.
- 5Sanity check against the rectangle around it
Draw the smallest rectangle that contains the whole plot and work out its area. Your total must be smaller than that and bigger than about half of it. Anything outside that range means a piece has been counted twice or dropped.
A curved boundary has no exact answer from a tape measure, so approximate it. Run a straight line that cuts off as much as it adds, or take offsets at even spacing along a baseline and treat each strip as a trapezoid. Both get you within a percent or two, which is closer than the boundary itself is defined.
Square footage, acres and the numbers that go with them
Square footage is just area measured in feet, and the arithmetic is the same length times width that every other rectangle uses. A 12 foot by 12 foot room is 144 square feet. A 16 foot by 12 foot bedroom is 192. Ten by ten is 100, fifteen by fifteen is 225, twenty by twenty is 400. Where it stops being obvious is the conversions, because area units do not scale the way length units do.
One foot is 12 inches, so it feels as though one square foot should be 12 square inches. It is 144, because both lengths are converted and the factor gets squared. The same trap turns 3 feet per yard into 9 square feet per square yard, and 100 centimetres per metre into 10,000 square centimetres per square metre. Every area conversion on this page is a length factor squared, which is also why a 1 percent measuring error becomes roughly a 2 percent error in the area.
Measure both sides 1 percent too long. Each measurement you use is 1.01 times the truth.
Area multiplies two lengths. So the area comes out 1.01 × 1.01 = 1.0201 times the truth, which is 2.01 percent too large.
Double your percentage error and you have the area error. On a 2,000 square foot house, measuring 1 percent long adds about 40 square feet that is not there.
Land units are where the arithmetic gets peculiar, because they were not designed, they accumulated. An acre is exactly 4,046.8564224 square metres, which is exactly 43,560 square feet and exactly 4,840 square yards. Those numbers look arbitrary until you learn the shape they came from: an acre is one chain by one furlong, a chain being 22 yards and a furlong 220, which is the strip a team of oxen could plough in a day. A hectare, invented much later and on purpose, is simply a square 100 metres on a side, so 10,000 square metres exactly, with 100 hectares to the square kilometre. One hectare is 2.47105 acres, so a hectare is the bigger of the two by a comfortable margin.
Ordering materials: flooring, turf and tiles are bought with a waste allowance on top of the measured area, conventionally about 10 percent, and more for diagonal layouts or rooms with many cuts. The area arithmetic here is exact. The offcuts are not.
Every formula on this page
Fourteen of these are in the picker above. The badge says how far each one gets you: a full answer, an area with no fixed perimeter, an approximation, or a method rather than a formula.
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle or square | l × w | 2(l + w) |
| Parallelogram | b × h | Needs the slanted side |
| Trapezoid | ½(b₁ + b₂) × h | Needs both slanted sides |
| Rhombus | ½ p q | 4√((p/2)² + (q/2)²) |
| Kite | ½ e f | Needs both side lengths |
| Triangle, base and height | ½ b h | Not fixed at all |
| Triangle, three sides | √(s(s−a)(s−b)(s−c)) | a + b + c |
| Circle | π r² | 2πr |
| Semicircle | ½ π r² | πr + 2r |
| Sector | (θ/360) π r² | 2r + (θ/360)2πr |
| Ellipse | π a b | Approximation only |
| Ring or annulus | π(R² − r²) | 2π(R + r) |
| Regular polygon | n s² ÷ (4 tan(π/n)) | n × s |
| L-shape | (L × W) − (l × w) | 2(L + W) |
| Irregular polygon | ½|∑(xₕyₖ₊₁ − yₕxₖ₊₁)| | Needs coordinates |
| Anything else | Cut it up and add | Measure the outline |
Area units, exactly
| Unit | In square metres | Note |
|---|---|---|
| Square metre | 1 | SI unit |
| Square inch | 0.00064516 | 144 to the square foot |
| Square foot | 0.09290304 | Exact, one ninth of a square yard |
| Square yard | 0.83612736 | Exactly 9 square feet |
| Are | 100 | A 10 metre square, a hundredth of a hectare |
| Acre | 4,046.8564224 | 43,560 ft², 4,840 yd² |
| Hectare | 10,000 | A 100 metre square, 2.47105 acres |
| Square kilometre | 1,000,000 | 100 hectares |
| Square mile | 2,589,988.110336 | Exactly 640 acres |
The acre is shipped here at its exact value, 4,046.8564224 square metres. The rounded 4,046.9 that most calculators publish is close enough for a field and wrong enough to notice on a survey. The United States survey acre is a third definition, 4,046.872 square metres, larger by about 24 square inches.
Area, from a tape measure to acres
Fourteen flat shapes, the perimeter each one does and does not fix, and the unit arithmetic that catches almost everybody out.
Area is the amount of flat ground or surface a shape covers, and because two lengths are multiplied together it is always measured in square units. This area calculator covers fourteen shapes in one picker, reports the answer in the unit you measured in, and then fills in square metres, square feet, square yards, acres and hectares from the same figure, so a bedroom and a paddock go through the same tool.
Most area questions are not really geometry questions. Somebody wants to know how much turf to order, whether a plot is worth its asking price, or how much paint covers a gable end. The formula is the easy middle step. The two steps either side, picking the shape at the start and getting the units right at the end, are where the mistakes live, so that is where this guide spends its time.
Why area units are squared, and what that costs you
Every area unit is a length unit multiplied by itself. A square metre is a square one metre on a side, a square foot is a square one foot on a side, and the SI unit of area is the square metre. The awkward part is that conversions between area units are the length factor squared, not the length factor. One foot is 12 inches, so a square foot is not 12 square inches, it is exactly 144. One yard is 3 feet, so a square yard is exactly 9 square feet, not 3. One metre is 100 centimetres, so a square metre is 10,000 square centimetres. Square anything that converts and you will not go wrong.
The same squaring applies to your mistakes. If both sides of a room are measured 1 percent too long, the area comes out 1.01 times 1.01, which is 1.0201, or about 2 percent too large. Double your percentage measuring error and you have your area error. On a 2,000 square foot house that is roughly 40 square feet of floor that does not exist, which is enough to change a flooring order.
Choose the shape before you choose the formula
Almost every area tool on the web opens a dropdown and assumes you already know which shape you are holding. In practice that is the hard part, and choosing wrongly costs far more than any rounding. A gable end is a triangle, but the number to enter is the vertical rise to the ridge, not the rafter length, because the rafter is the slope. A drive that widens as it leaves the road is a trapezoid, not a rectangle. A path running round a pond is a ring, which is one circle minus another and far easier than treating it as a bent rectangle. A lounge with a chimney breast is a rectangle with a cutout. The decision table higher up this page maps these and more, and it is worth reading before you type any numbers.
Perimeter is a separate question, and sometimes it has no answer
Area tells you how much ground; perimeter tells you how much edge. They are different questions and they often have different buyers: square footage for the carpet, perimeter for the skirting board; area for the turf, perimeter for the edging. This page prints both, which almost no shape based calculator does.
It also refuses to print a perimeter when there genuinely is not one. A triangle described by a base and a height has no fixed perimeter, because you can slide the apex sideways as far as you like without changing either number, and the two other sides change every time. A trapezoid given two bases and a height is the same story, and so is a parallelogram given a base and a height. In each case the calculator says so and tells you which extra measurement closes the shape. A guess dressed up as a number would be worse than an honest gap.
Square footage, acres and hectares
Square footage is simply area measured in feet. A 12 foot by 12 foot room is 144 square feet, a 16 foot by 12 foot bedroom is 192, and ten by ten, fifteen by fifteen and twenty by twenty give 100, 225 and 400. Nothing clever is happening; the arithmetic is length times width.
Land units are stranger, because they were not designed so much as inherited. An acre is exactly 4,046.8564224 square metres, which is exactly 43,560 square feet and exactly 4,840 square yards. Those numbers look arbitrary until you see the shape behind them: an acre is one chain by one furlong, a chain being 22 yards and a furlong 220, which is about the strip a team of oxen could plough in a day. A hectare was invented later and on purpose, so it is simply a square 100 metres on a side, exactly 10,000 square metres, with 100 hectares to the square kilometre. One hectare is 2.47105 acres, so the hectare is the larger of the two by a comfortable margin. The calculator ships the exact acre rather than the rounded 4,046.9 that most competitors publish.
When the ground refuses to be a shape
Real plots are rarely one clean outline. The surveyor's answer is the shoelace formula, which turns a list of corner coordinates into an area and has been used for two centuries, but it needs coordinates and nobody standing in a field with a tape measure has any. The practical answer is decomposition. Sketch the plot, slice it with straight lines into rectangles and triangles, measure each piece fully, run each through the calculator in the same unit, then add the pieces and subtract the holes. A pond, a patio you are not turfing and a doorway in a wall are all holes.
Check the total against the smallest rectangle that would contain the whole plot. Your answer has to be smaller than that rectangle and bigger than roughly half of it. Anything outside that range means a piece was counted twice or dropped. A curved boundary has no exact answer from a tape measure at all, so run a straight line that cuts off about as much as it adds, or take offsets at even spacing along a baseline and treat each strip as a trapezoid. Either gets you within a percent or two, which is usually closer than the boundary itself is defined.
Three jobs worked through
A lounge with a chimney breast. The room is 20 ft by 15 ft, and the breast takes out 8 ft by 6 ft. That is 300 minus 48, so 252 square feet of floor. Skirting board follows the perimeter, which is still the 70 feet of the rectangle around it, because the breast adds back exactly what it cuts away.
A plot paced out in metres. A rectangle 30 m by 45 m is 1,350 square metres. That is 0.135 hectares and 0.334 acres, or just over a third of an acre, which is the figure an estate agent will quote. The same plot is 14,531 square feet.
Forty metres of fencing. Bend it into a circle and it encloses 127 square metres. A regular hexagon gives 115, a 10 metre square gives 100, and an 18 by 2 strip gives 36. Same fence, same money, a quarter of the ground. Shape is not decoration when you are buying an edge.
Measuring so the answer is worth having
- 1One unit, all the way through
Convert every measurement before you start, not halfway. Mixing feet with inches, or metres with centimetres, is the single most common failure in this subject.
- 2Radius, not the distance across
Round shapes take the radius. If the tape went across the whole circle that is the diameter, so halve it first, or the area comes out four times too big.
- 3Height means at right angles
For a triangle, a trapezoid or a parallelogram the height is the perpendicular drop, never the slanted side. The slanted side is always longer, so using it inflates the answer.
- 4Measure the surface, then add for waste
Flooring, turf and tiles are bought with an allowance on top, conventionally about 10 percent and more for diagonal layouts or rooms full of cuts. The arithmetic is exact; the offcuts are not.
Frequently Asked Questions
How do I work out the square footage of a room that is not a plain rectangle?
Split it. A lounge with a chimney breast is a rectangle with a rectangle taken out, so use the L-shape mode: overall length and width first, then the cutout. A room with a bay is a rectangle plus a second shape, so work out both and add them. Anything more complicated is the same idea repeated, and the running total is what you order from. Keep every measurement in feet while you do it, because converting halfway through is what usually goes wrong.
Why is one square foot 144 square inches rather than 12?
Because both sides get converted. A square foot is a square one foot on each side, which is 12 inches by 12 inches, and 12 times 12 is 144. The rule holds for every area conversion: take the length factor and square it. Three feet to the yard becomes 9 square feet to the square yard. One hundred centimetres to the metre becomes 10,000 square centimetres to the square metre. Anybody who converts an area with the plain length factor is out by that factor all over again.
How do I turn a measurement taken in feet into acres or hectares?
Divide. One acre is exactly 43,560 square feet, so a 20,000 square foot plot is 20,000 divided by 43,560, which is 0.459 acres. For hectares, convert to square metres first by multiplying square feet by 0.09290304, then divide by 10,000, since a hectare is exactly 10,000 square metres. The calculator above does both from whatever unit you measured in, so a plot paced out in feet comes back in acres without a second tool.
Is an acre bigger than a hectare, and where is each one used?
The hectare is bigger, and by a wide margin: one hectare is 2.47105 acres, so it takes nearly two and a half acres to make one. The acre is an inherited unit, exactly one chain by one furlong, and it is still standard for land in the United States, the United Kingdom and much of the Commonwealth. The hectare was defined deliberately as a square 100 metres on a side and is the metric standard for farmland and forestry almost everywhere else. Property listings in Britain often print both, which is why the two keep turning up side by side.
What is the difference between area and perimeter?
Area is the ground covered, measured in square units. Perimeter is the distance round the outside, measured in plain length units. They answer different purchases: area buys the carpet, the turf and the paint, while perimeter buys the skirting, the edging and the fence. They also move independently, which surprises people. Two plots of exactly one acre can need 225 metres of fence or more than 800, depending only on the shape, and the comparison band on this page shows that with numbers.
Why will the calculator not give me a perimeter for every shape?
Because for some inputs no perimeter exists. A triangle described only by a base and a height can have its apex slid sideways forever without either number changing, and the other two sides change every time, so there is no single answer to give. The same is true of a trapezoid given two bases and a height, and of a parallelogram given a base and a height. In each case the tool says which extra measurement closes the shape: three sides for the triangle, both slanted sides for the trapezoid, one slanted side for the parallelogram. An ellipse is a different case again, since its perimeter has no exact formula at all and the figure shown is Ramanujan's approximation.
Should I measure the sloping side of a gable or the height straight up?
Straight up. Every area formula that mentions a height wants the perpendicular distance, at right angles to the base, and on a gable end that is the vertical rise from the eaves line to the ridge. The rafter is the sloping side and it is always longer, so using it makes the wall look bigger than it is and you buy paint or render you do not need. The same rule catches people on parallelograms and trapezoids: the slanted side is not the height.
What shape gives the most area for a fixed length of fencing?
A circle, always, and the gap is larger than people expect. Forty metres of fencing encloses 127 square metres as a circle, 115 as a regular hexagon, 100 as a ten metre square, 77 as an equilateral triangle and only 36 as an 18 by 2 strip. Among rectangles the square is the best you can do, and the more lopsided the rectangle the worse it gets. The practical version of that rule is simple: if you are paying by the metre for an edge, keep the shape as round and as compact as the site allows.
How do I deal with a plot whose boundary curves?
Approximate it, because a curve has no exact answer from a tape measure anyway. The quick method is to draw a straight line across the curve that cuts off about as much ground as it adds, then treat the plot as the straight sided shape that remains. The careful method is offsets: run a straight baseline along the plot, measure the distance out to the boundary at even spacing, and treat each strip between two measurements as a trapezoid. Both land within a percent or two, which is normally closer than the boundary is actually defined on the deed.
How much does a small measuring error change the area?
Roughly twice as much as the error itself, because area multiplies two lengths. Measure both sides of a room 1 percent long and the area comes out 1.01 times 1.01, which is 1.0201, or about 2 percent too large. Get one side wrong and only that side is affected, so the area is out by the same 1 percent. The working rule is to double your percentage error for an area, and on a 2,000 square foot house a 1 percent overshoot is about 40 square feet that is not there.