Area Calculator

Area is the size of a two-dimensional figure in a plane. The everyday way to picture it is the amount of paint needed to cover a surface. It is the two-dimensional counterpart of length in one dimension and volume in three. The SI unit of area is the square meter, m², and the equations below cover the seven shapes that come up most often.

Every one of them reduces to the same idea. A shape is measured by how many unit squares fit inside it, and each formula is a shortcut for counting those squares without actually laying them out.

Rectangle

A rectangle is a quadrilateral with four right angles. It is one of the simplest shapes, and its area needs only the length and width. A quadrilateral by definition is a polygon with four edges and vertices. Length usually refers to the longer pair of edges and width to the shorter pair. When the two are equal the shape is a square, which is a special case of a rectangle rather than a different thing.

A worked example in imperial units. A fenced plot measures 220 feet by 99 feet, giving 220 × 99 = 21,780 square feet. That number is worth recognising: an acre is 43,560 square feet, so the plot is exactly half an acre. The acre itself is defined as one chain by one furlong, which is 66 feet by 660 feet, and the arithmetic works out to the same 43,560.

The foot has been exactly 0.3048 meters since the international yard and pound agreement of 1959. Before that it drifted, because for most of history units of length were based on the human body, which is not consistent from one person or century to the next. That inconsistency is the reason SI exists.

Triangle

There are many equations for the area of a triangle depending on what you know. The familiar one is half the base times the height. The calculator above uses a different one, because it needs only the three side lengths and no angle or height at all: Heron's formula, named for Hero of Alexandria, a Greek mathematician and engineer of the first century.

where s is the semiperimeter:

Take an equilateral triangle with sides of 77 feet. The semiperimeter is (77 + 77 + 77) / 2 = 115.5, and the area is the square root of 115.5 × 38.5 × 38.5 × 38.5, which is 2,567.33 square feet.

Two practical notes. Any two sides must add to more than the third, or no triangle exists and the formula returns the square root of a negative number; the calculator checks this and says so rather than printing an error. And Heron's formula loses accuracy on very thin triangles, where the subtractions cancel most of the significant digits. Numerical libraries use a sorted, rearranged version for that case, which is unlikely to matter for a garden or a roof but does matter in surveying software.

Trapezoid

A trapezoid is a simple convex quadrilateral with at least one pair of parallel sides. Convex means no interior angle exceeds 180°, and simple means the shape does not intersect itself, so two non-adjacent sides never cross. The parallel sides are the bases and the other two are the legs. Several sub-classifications exist, and the area is found the same way for all of them.

where b₁ and b₂ are the bases and h is the height, meaning the perpendicular distance between them rather than the length of a leg.

A BMX ramp with a trapezoidal side face, 9 feet high, with a bottom base of 29.528 feet and a top base of 9 feet, has a face area of (9 + 29.528) / 2 × 9 = 173.376 square feet.

The formula is really an average in disguise. Averaging the two bases gives the width of a rectangle with the same area and the same height, which is why a trapezoid with equal bases collapses neatly into the rectangle formula.

Circle

A circle is the set of all points in a plane at a fixed distance from a centre point. That distance is the radius. Everything about a circle follows from the radius and the constant π, the ratio of any circle's circumference to its diameter.

A crop circle with an outer radius of 15 feet needs π × 15² = 706.858 square feet of flattened crop.

π is irrational, so its decimal expansion never ends and never repeats, and it is also transcendental, proven by Ferdinand von Lindemann in 1882. That second property settled the ancient problem of squaring the circle: constructing a square of equal area with compass and straightedge alone is impossible. Archimedes pinned π between 223/71 and 22/7 by comparing 96-sided polygons inscribed in and circumscribed around a circle, which is the same idea a modern calculator uses, only with a great deal more patience.

Sector

A sector is the slice of a circle enclosed by two radii and the arc between them. Its area is the whole circle scaled by the fraction of the full turn that the angle covers.

or, when the angle is in radians rather than degrees:

A blackberry pie of radius 16 inches, cut so each person gets 60° of it, gives each portion 60/360 × π × 16² = 134.041 square inches.

The radian version is tidier because a radian is defined as the angle subtending an arc equal in length to the radius, so a full turn is 2π radians and the fraction cancels neatly. This calculator takes degrees, since that is what most people measure in, and converts internally.

Ellipse

An ellipse is the generalised form of a circle: the curve traced by every point whose distances to two fixed points, the foci, add to the same total. When the two foci coincide, the ellipse is a circle. The semi-major axis is the longest radius and the semi-minor axis the shortest, and the words major and minor refer to the full diameters rather than these halves.

where a and b are the semi-major and semi-minor axes. The equation is the circle formula with two different radii instead of one, which is exactly what an ellipse is.

A scale model of Earth's orbit measuring 18 feet by 20 feet covers π × 18 × 20 = 1,130.97 square feet.

The area is straightforward; the perimeter is famously not. No formula in elementary functions gives the circumference of an ellipse, only an elliptic integral, which is where that family of functions got its name. Ramanujan's 1914 approximation is the one still used in practice and is accurate to a few parts per billion for moderate shapes. It is a good reminder that a shape can be simple to measure one way and genuinely hard another.

Parallelogram

A parallelogram is a simple quadrilateral with two pairs of parallel sides, where opposite sides and opposite angles are equal. Rectangles, rhombuses and squares are all special cases of it. A parallelogram can be cut into a right triangle and a trapezoid and rearranged into a rectangle, which is why its area formula is the same as a rectangle's, with base and height in place of length and width.

The height is the length of the perpendicular between a pair of bases, not the slanted side. This is the single most common mistake with this shape: measuring along the slope inflates the answer, and the more the shape leans the worse the error gets.

One rhomboidal face of a crystal measuring 20 feet by 18 feet has an area of 360 square feet.

The underlying principle is Cavalieri's: two shapes of equal height whose horizontal slices match in length at every level have the same area. Push the top of a rectangle sideways and it becomes a parallelogram, with every slice the same length as before, so the area never changes no matter how far it leans.

Why the formulas look the way they do

All seven equations descend from one idea, and seeing the chain makes them much harder to forget.

The rectangle is the base case. Laying out unit squares in a grid of l columns and w rows gives l × w squares, and no derivation is needed, because the shape is the definition of area made visible.

The parallelogram comes next. Slice the triangle off one end and slide it round to the other, and you have a rectangle of the same base and the same height. Nothing was added or removed, so the area is unchanged, and b × h follows immediately.

The triangle is half of that. Take any triangle, copy it, rotate the copy 180° and join the two along a shared edge: the result is a parallelogram with the same base and height. One triangle is therefore half the base times the height. The Heron version reaches the same area without knowing the height, which is why it looks so different and agrees anyway.

The trapezoid is built the same way. Copy it, rotate the copy, and the pair fits together into a parallelogram whose base is b₁ + b₂ and whose height is h. Half of that is the trapezoid, which is exactly the averaging formula above.

The circle takes the nicest argument of all. Cut it into a large number of thin sectors and lay them alternately point-up and point-down, like the teeth of a zip. The result approaches a rectangle whose height is the radius and whose width is half the circumference, since half the arcs sit along the top and half along the bottom. That rectangle has area r × πr, which is πr². Make the slices thinner and the approximation improves without limit, which is the argument Archimedes made and, in modern language, an integral.

The sector is a fraction of that circle, and the ellipse is a circle stretched by different factors in two directions, which multiplies the area by both of them. Seven formulas, one idea, and a chain you can rebuild from scratch when memory fails.

Shapes that are not on this list

Most real outlines are not textbook shapes. Two approaches cover nearly everything. The first is decomposition: split the outline into rectangles and triangles, work out each, and add them up. A room with a bay window is a rectangle plus a trapezoid, and a gable wall is a rectangle plus a triangle.

The second is the surveyor's formula, also called the shoelace formula, which gives the exact area of any simple polygon from the coordinates of its corners. List the vertices in order around the shape, multiply each x by the next y, subtract each y times the next x, add the lot and halve the absolute value. It gets its nickname from the way the multiplications cross over when written out in two columns. Every mapping and CAD package uses it, and it needs no angles, no heights and no assumptions about the shape beyond the edges not crossing.

For genuinely curved outlines such as a lake, the practical methods are counting squares on a grid overlay or letting software integrate the traced boundary, which is the same principle at a finer resolution.

Area units and why they are awkward

Unit	Area in m²
square meter	SI unit
hectare	10,000
square kilometer (km²)	1,000,000
square foot	0.0929
square yard	0.8361
acre	4,046.9 (43,560 square feet)
square mile	2,589,988 (640 acres)

Notice how uneven the imperial column is. Those numbers are irregular because the units were defined by different things at different times: the acre began as the area a yoke of oxen could plough in a day, and the chain and furlong that formalised it were agricultural measures long before they were legal ones. The metric column is uniform by design, which is the whole point of a decimal system.

One conversion trap catches people constantly. Area units do not scale like length units. A yard is 3 feet, but a square yard is 9 square feet, not 3. A kilometer is 1,000 meters, and a square kilometer is 1,000,000 square meters. Whenever you convert an area, the length factor gets squared.

The same rule explains why doubling a room's dimensions quadruples the flooring, and why a pizza of twice the diameter is four times the pizza. Area grows with the square of length, which is one of the most useful pieces of intuition in practical measurement and one of the least intuitive until you have been caught by it once.

Using the calculator

Each shape takes its own measurements, and every input carries its own unit menu, so you can mix feet and inches without converting first. The answer is reported in the square of the first input's unit, with the working shown underneath and your own numbers substituted into the formula. "Show result in other units" converts the same answer into square feet, square yards, acres, hectares, square kilometers and square miles at once, which is the step most often needed for anything to do with land or materials.