Area Calculator

Choose a shape, type in its dimensions, and the area appears instantly to four decimal places. The calculator handles the five shapes behind almost every practical area question — rectangle, square, triangle, circle and trapezoid. Fields you do not need stay hidden: a square asks for one number, a trapezoid for three. The result is always in the square of whatever unit you type, so meters in gives square meters out and feet in gives square feet.

What each dimension means

The letters a, b and h stand for different measurements depending on the shape, so check this table before typing:

Note that the circle takes the radius, not the diameter. If you measured across the full width, halve it before entering.

The five formulas

A = a × b for a rectangle, and A = a² for a square

A = a × b ÷ 2 for a triangle

A = πa² for a circle

A = (a + b) ÷ 2 × h for a trapezoid

The rectangle is the reference case: area counts how many unit squares tile the shape, and a grid a units long by b units wide holds exactly a × b of them. Every other formula reduces to that idea. A square is a rectangle with equal sides, so its area is a × a. A triangle is half of the rectangle built on the same base and perpendicular height, which is why those two lengths are multiplied and then halved. A trapezoid averages its two parallel sides and multiplies by the height, treating the sloped shape as a rectangle whose width is the mean of the top and bottom. The circle stands apart, because no whole number of straight-sided squares ever tiles it exactly.

With the default values — a rectangle with a = 4.5 and b = 3.2 — the area is 4.5 × 3.2 = 14.4. Switch to triangle with the same two numbers and the result halves to 7.2, since a triangle covers half of the rectangle built on the same base and height. Keep a = 4.5 and pick circle instead, and the number jumps to π × 4.5² = 63.6173, because a now describes a radius sweeping a far larger region.

Where the formulas come from

Area is one of the oldest problems in mathematics, and it began as a problem about land. The Greek word geometria means earth-measurement, and in Book II of his Histories the fifth-century BCE writer Herodotus credited the subject to Egyptian officials who re-measured farmland after the Nile's annual flood so that each holding paid tax in proportion to what was left of it. The scribes already used most of the formulas on this page. The Rhind Mathematical Papyrus, copied around 1550 BCE by a scribe named Ahmes from a text roughly three centuries older written in the reign of Amenemhat III, computes a rectangle as length times width, a triangle as half the base times the height, and a trapezoid from its two parallel sides and the height. Its problem 50 takes a circle as the square of eight-ninths of the diameter, which is equivalent to using 256/81, about 3.1605, for π — roughly 0.6 percent above the true value, and remarkable for the period.

Old Babylonian clay tablets from the same broad era, the centuries between 2000 and 1600 BCE, used a rougher circle, measuring its area as one-twelfth the square of the circumference, which amounts to setting π equal to three. A tablet found near Susa does better, implying 25/8, or 3.125. The move from recipe to proof came from the Greeks. In the fifth century BCE Hippocrates of Chios achieved the first known quadrature of a curved figure, squaring certain crescent-shaped lunes and so proving that a region with curved edges could equal a straight-sided one exactly. Eudoxus of Cnidus then made the method of exhaustion rigorous, approximating a curved region ever more closely with inscribed polygons. Around 300 BCE Euclid's Elements gathered this work into an axiomatic system: its plane books show how straight-sided areas can be compared and transformed into one another, and Book XII uses exhaustion to prove that circles are to one another as the squares on their diameters.

Archimedes brought the circle under control. In Measurement of a Circle, written about 250 BCE, he proved that a circle has the same area as a right triangle whose two shorter sides equal the radius and the circumference — the geometric core of A = πr². By inscribing and circumscribing regular 96-sided polygons he pinned π between 223/71 and 22/7, roughly 3.1408 and 3.1429. His polygon-squeezing is the direct ancestor of the integral calculus that Newton and Leibniz built in the late seventeenth century, which finally gave a general way to find the area under any curve rather than the handful of shapes the ancients could handle by hand.

Why every area comes out squared

The calculator does not care what unit you measure in, but the result is always that unit squared: meters in, square meters out. Two consequences follow. First, never mix units within one shape — a in meters and b in centimeters produces a meaningless number. Second, converting an area between systems uses the square of the length factor, not the factor itself. One meter is about 3.28084 feet, yet one square meter is 3.28084² = 10.7639 square feet. The same effect turns a doubling of every side into a quadrupling of area, which is exactly why a circle entered with its diameter instead of its radius comes out four times too large. The area converter on this site handles the unit step for you.

Units and names around the world

Naming and units diverge by market, and the gaps catch people out. The four-sided shape this calculator calls a trapezoid — a quadrilateral with exactly one pair of parallel sides — is a trapezium in the UK, Ireland and Australia, where trapezoid instead means a quadrilateral with no parallel sides at all. Pairing a British textbook with an American tool is a reliable way to reach for the wrong word. US home listings quote square feet, while UK and EU listings use square meters. Land carries its own units with long histories. The acre began in medieval England as roughly the area a team of oxen could plough in a day, and an English statute of about 1300 fixed it at forty perches by four — one furlong by one chain, or 66 by 660 feet, which is 43,560 square feet. The hectare comes from the metric system introduced in France in the 1790s: the law of 18 Germinal, Year III, dated 7 April 1795, defined the are as 100 square meters, and a hectare is 100 ares, or 10,000 square meters. Land is commonly priced per acre in the US and UK and per hectare across most of the EU and Australia.

What this covers and what it does not

All five formulas assume flat shapes and perpendicular measurements. The triangle height must meet the base at a right angle, and the trapezoid height is the straight-line distance between the two parallel sides, not the length of a slanted leg. For an irregular outline — an L-shaped room, a garden with a corner cut off — split it into rectangles and triangles, run the calculator once per piece, and add the results. Curved or sloping surfaces such as a pitched roof, a dome or a hillside need their own geometry, and for those the ancient trick of cutting a region into ever-smaller pieces, later formalized by calculus, is still the idea underneath.

Frequently asked questions

How do I find the area of a circle if I know the diameter?

Halve the diameter to get the radius, then use the circle option. A circle 10 cm across has a radius of 5 cm and an area of π × 5² = 78.5398 cm². Entering the diameter directly is the most common circle mistake and inflates the answer fourfold.

Is a trapezoid the same as a trapezium?

In American English a trapezoid is a quadrilateral with exactly one pair of parallel sides, and that is the shape this calculator computes with A = (a + b) ÷ 2 × h. In British English the very same shape is called a trapezium. The two words exist in both dialects with swapped meanings, so go by the formula rather than the name.

Which height do I use for a triangle?

The perpendicular height from the base to the opposite corner, not the length of a slanted side. In a right triangle the two legs serve directly as base and height. A triangle with base 6 and perpendicular height 4 has an area of 12 no matter how lopsided it looks.

How do I work out the floor area of a room?

Measure the length and width at floor level and use the rectangle option. A 12 ft by 14 ft bedroom is 168 sq ft, and a 3.6 m by 4.2 m room is 15.12 m². For an L-shaped room, split it into two rectangles and add the two results.