Area Calculator — Every Shape, Every Unit, Instantly

Area is the two-dimensional size of a surface — how many 1 × 1 unit squares fit inside it. Each shape has its own formula, but they all answer that same question:

Every area question — a bedroom floor, a garden bed, a plot of land, the area of a shape on a homework page — is answered the same way. Area asks one simple thing: how much flat surface sits inside the boundary? Work through these four steps and any shape below gives up its number.

1. Name the shape. Decide what you are actually looking at. A room is a rectangle, a round patio is a circle, a gable end is a triangle, a sloped plot is often a trapezoid. The eight shapes in the tool cover nearly everything you will meet in a house, a garden or a classroom — and anything odder can be split into pieces that are on the list (the irregular-shapes section below shows how).

2. Measure every length in the same unit. Mixing feet and inches inside one formula is the most common cause of wrong areas. Either convert first — 6 inches is 0.5 ft — or enter each measurement with its own unit in the tool and let it convert for you.

3. Apply that shape’s formula. Each shape has one, and the master table in the next section lists them all with the letters decoded. There is no single formula that fits every outline; the shape always chooses the formula.

4. Report the answer in square units. An answer of “108” means nothing until it says 108 square feet. Area is always squared — square inches, square feet, square meters, acres — because it counts two-dimensional space.

The picture worth keeping in mind is a floor tile. One square foot is exactly one 1 ft × 1 ft tile, and finding an area is counting how many of those tiles cover the surface — whole tiles plus the fractions along the edges. Every formula below is just a shortcut for that count: multiplying length by width counts the tiles in rows instead of one by one. That is also all the area calculator above does — it runs the same count instantly, in whichever of its eight output units you ask for.

One scale note before you start: pick the unit that fits the job. Square inches suit a phone screen or a sheet of paper; square feet and square yards suit rooms and flooring; square meters suit metric listings; acres and hectares suit land; square miles and square kilometers suit whole districts. A farm measured in square inches is technically correct and practically useless — the right unit keeps the number readable at a glance.

Read next: for land itself — lots, parcels and acreage — the companion guide is Acreage and Lot Size.

There is no single formula for area — there is one per shape, because each outline fills its unit squares in a different pattern. This table is the whole toolkit on one screen. Find your shape, note what each letter stands for, and check your arithmetic against the worked example before you trust a number with a real order of flooring or turf.

Two reading notes. First, the sector row and the tool agree: the calculator takes the angle directly, and a quarter slice (90°) of a circle with radius 10 ft comes to exactly a quarter of the full circle — 78.54 sq ft, the same figure as the circle row with radius 5 ft, because doubling the radius quadruples the area and a quarter of that lands back on the same number. Second, everything here is flat, two-dimensional measurement. If you are measuring the outside of a solid — a box, a tank, a sphere — that is surface area, a different quantity with its own formulas; the Surface Area Calculator handles those.

The rectangle area formula is the first area formula anyone learns, and it earns that place: the area of a rectangle is its length multiplied by its width. The multiplication is not a rule to memorise blindly — it is a counting shortcut. Lay 1 ft × 1 ft tiles across a floor and they form rows: a room 12 ft long fits 12 tiles along its length, and at 9 ft wide it fits 9 rows. Counting tile by tile gives 108 tiles; multiplying 12 × 9 gives the same 108 in one step. Every rectangle area on earth is that product, which is why “rectangle area” and “length times width” are the same sentence said two ways.

Worked example. A bedroom measures 12 ft by 9 ft. Area = 12 × 9 = 108 sq ft. That single number drives the real decisions: flooring is ordered by it (plus a waste allowance), a rug is sized against it, and a paint estimate starts from it. For project pricing on top of the geometry, the Square Footage Calculator adds L-shaped rooms and a cost estimator to the same math.

Running it backwards. The formula also solves for a missing side: if a rectangular plot has an area of 96 sq ft and a width of 8 ft, its length is 96 ÷ 8 = 12 ft. Surveyors and gardeners use this direction constantly — the area is on the deed, one fence line is known, and the other side is the unknown. The area-equations section below collects more of these reverse problems.

The square: a rectangle whose sides match

The area of a square is the rectangle formula with length equal to width, so it collapses to A = s² — side times itself. A square tile, plot or patio with a side of 7 m has an area of 7 × 7 = 49 sq m. The square also has a diagonal shortcut worth knowing: A = d² ÷ 2, where d is the corner-to-corner diagonal. A square whose diagonal measures 10 ft has an area of 100 ÷ 2 = 50 sq ft — handy when the diagonal is the one measurement you can actually take, across a planted bed or a crowded room.

Read next: measuring a whole room from first tape reading to paint shelf — alcove, walls and waste allowance included — is its own worked guide: How to Find the Area of a Room.

A circle’s area is A = π × r²: square the radius, then multiply by π. The one mistake that ruins more circle answers than any other is feeding the formula the diameter. The radius is half the diameter, so halve first, always. A round rug with a radius of 5 in covers π × 25 = 78.54 sq in; a circular patio sold as “10 ft across” has a radius of 5 ft and covers π × 25 = 78.54 sq ft — same digits, different unit, because the radius is the same. The calculator keeps π at full precision through the working and rounds only the display, so its answers will match these to the last digit. For diameter, circumference and arc questions around the same circle, the Circle Calculator is the companion tool.

The ellipse — the shape behind every “area of oval” search

An oval is an ellipse, and its formula is the circle’s closest relative: A = π × a × b, where a and b are the semi-axes — half the full width and half the full height, measured from the centre. An oval lawn 12 ft wide and 8 ft tall has a = 6 ft and b = 4 ft, so its area is π × 6 × 4 = π × 24 = 75.40 sq ft. The trap mirrors the circle’s: enter the full width and height instead of the semi-axes and the answer comes out exactly four times too big. Halve both measurements first, then multiply — that is the whole of the area of oval shapes, and it is why the tool asks for semi-axis a and semi-axis b rather than width and height.

Triangle: A = ½ × b × h. The half is not arbitrary. Take any triangle, put a second copy of it against the first, rotated 180°, and the pair forms a parallelogram whose area is base × height — so one triangle is exactly half of that. A triangular gable with a base of 10 ft and a height of 6 ft has an area of ½ × 10 × 6 = 30 sq ft. When you know the three sides instead of a height, the tool’s triangle mode switches to Heron’s formula: sides of 5, 12 and 13 give a semi-perimeter of 15 and an area of √(15 × 10 × 3 × 2) = √900 = 30 sq units — the same answer the right-triangle shortcut gives (½ × 5 × 12), because 5-12-13 is a right triangle. For sides, angles and perimeter work beyond area, use the Triangle Calculator.

Trapezoid: A = ½ × (a + b) × h. A trapezoid has two parallel sides of different lengths, so the formula averages them and multiplies by the height — it is the area of a rectangle whose width is the average of the two. With parallel sides of 8 ft and 14 ft and a height of 5 ft: ½ × (8 + 14) × 5 = 11 × 5 = 55 sq ft. Sloped garden plots and tapered driveways are the everyday homes of this formula.

Parallelogram: A = b × h. That is the whole trick behind every parallelogram area calculator result — but the h is the perpendicular height: the vertical distance between the two bases, not the length of the slanted side. A parallelogram with a base of 9 ft and a perpendicular height of 5 ft has an area of 45 sq ft, even if its slanted side measures 6 ft. Use the slant and you would compute 54 sq ft — a 20% overstatement from one wrong measurement. Slide a pair of scissors through the shape mentally: cut the triangle off one end, move it to the other, and the parallelogram becomes a rectangle with the same base and the perpendicular height. That picture is the proof.

Read next: triangles alone — Heron’s formula, the two-method cross-checks and the mistakes to skip — get a full guide: Area of a Triangle.

For a regular polygon — all sides equal, all angles equal — the standard area of polygon formula is A = ½ × perimeter × apothem. The apothem is the distance from the centre of the shape to the middle of one side, and the formula works because a regular polygon is really a fan of identical triangles, each with its base on one side and its tip at the centre. A regular hexagon with sides of 4 units has a perimeter of 24 and an apothem of about 3.46, so its area is ½ × 24 × 3.46 = 41.52 sq units. Hexagonal pavers, pentagonal planters and octagonal gazebos all yield to the same one line of arithmetic.

Four-sided shapes are less obedient. There is no single formula for a general quadrilateral — which is exactly why “area of quadrilateral calculator” searches end at methods rather than formulas. The method that always works: split the quadrilateral along one diagonal into two triangles, find each triangle’s area with ½ × base × height (the diagonal is the shared base; each height is measured perpendicular to it, from the opposite corner), and add. Worked example: a quadrilateral whose diagonal is 10 ft, with the two opposite corners sitting 4 ft and 3 ft from that diagonal, splits into triangles of ½ × 10 × 4 = 20 sq ft and ½ × 10 × 3 = 15 sq ft — a total area of 35 sq ft. Irregular lots with four crooked sides are measured this way in real surveying, one diagonal at a time.

Most real outlines — L-shaped rooms, patios with a bite taken out, garden beds that wrap around a corner — match no single formula. The method that handles all of them is split-and-add: cut the outline into rectangles, triangles and circles, run each piece through its own formula, and add the pieces. Its twin is subtract-the-cutout: draw the smallest rectangle that contains the shape, then subtract the parts of that box the shape does not cover.

Worked example, both ways. An L-shaped room fits inside a 12 ft × 10 ft box, with a 4 ft × 3 ft corner missing. Subtract: the box is 120 sq ft, the missing corner is 12 sq ft, so the room is 120 − 12 = 108 sq ft. Split instead: a 12 ft × 7 ft strip along the bottom (84 sq ft) plus an 8 ft × 3 ft piece above it (24 sq ft) gives 84 + 24 = 108 sq ft again. Two routes, one answer — that agreement is your check that the split was drawn correctly. Used this way, the tool above becomes your irregular shape area calculator: enter each rectangular or triangular piece as its own shape, note each result, and add them at the end. Our supporting guide on irregular shapes takes the method further, through curved cut-outs and grid counting.

A third route exists for outlines too crooked to split cleanly: lay a grid over the shape — graph paper, or a mental grid of 1 ft squares — count the whole squares inside it, then count the part-squares along the edge as halves and add them on. It gives an estimate rather than an exact area, but for a winding garden bed it is often the only practical method, and it is the same counting idea every formula on this page is a shortcut for.

Read next: the complete method set, including curved cut-outs and the coordinate check, is in Area of Irregular Shapes.

An area equation is just a formula with the unknown in a different seat. Because every area formula is built from multiplications, each one runs backwards with division (and, for the circle, a square root):

• Rectangle: area 96 sq ft, width 8 ft → length = 96 ÷ 8 = 12 ft.
• Triangle: area 30 sq ft, base 10 ft → height = 2 × 30 ÷ 10 = 6 ft (the ½ in the formula means the reverse step doubles first).
• Circle: area 78.54 sq ft → radius = √(78.54 ÷ π) = √25 = 5 ft.

As for the measurements themselves: every dimension box in the tool carries its own unit menu — feet, inches, meters, centimeters, yards, kilometers or miles — which makes it a small dimension calculator in its own right. Measure a patio as 5 ft one way and 30 in the other and you can type both exactly as measured; each dimension is converted into a shared unit before the formula multiplies them. That order matters. Converting after squaring is the classic blunder — a length in feet times a length in inches is not square anything until both speak the same unit. Enter dimensions as you measured them, and the normalising is done for you, in the right order, every time.

Area units convert by fixed factors — but the factors are the squares of the length factors, which is why they surprise people: doubling 12 inches to a foot is easy, but a square foot holds 144 square inches, not 12. These are the exact relations behind the tool’s eight output units.

Two conversions, worked. A 500 sq ft apartment, for a listing written in metric: 500 × 0.092903 = 46.45 sq m. And a 2-acre plot, for a fencing estimate in feet: 2 × 43,560 = 87,120 sq ft. The calculator performs both directions instantly — enter any shape, then switch the output unit — and our supporting guide on area units walks through the full table, including why the “square the factor” rule catches so many people out.

Which unit should you report? Match the unit to the trade you are talking to. Flooring and paint in the US are quoted in square feet, fabric and carpet often in square yards; European listings speak square meters; land is traded in acres in the US and hectares almost everywhere else; seed, turf and fertilizer coverage is printed per 1,000 sq ft or per acre. Converting into your listener’s unit before you quote prevents the most expensive misunderstanding in measuring — two people picturing areas a hundred times apart while using the same digits.

Read next: the full conversion table with the wrong-direction errors worked out is in Area Units and Conversions.