Area in the context of "Signed area"

In photography, exposure is the amount of light per unit area reaching a frame of photographic film or the surface of an electronic image sensor. It is determined by exposure time, lens f-number, and scene luminance. Exposure is measured in units of lux-seconds (symbol lx⋅s), and can be computed from exposure value (EV) and scene luminance in a specified region.

An "exposure" is a single shutter cycle. For example, a long exposure refers to a single, long shutter cycle to gather enough dim light, whereas a multiple exposure involves a series of shutter cycles, effectively layering a series of photographs in one image. The accumulated photometric exposure (H_v) is the same so long as the total exposure time is the same.

The square kilometre (square kilometer in American spelling; symbol: km) is a multiple of the square metre, the SI unit of area or surface area. In the SI unit of area (m), 1 km is equal to 1M(m).

The hectare (/ˈhɛktɛər, -tɑːr/; SI symbol: ha) is a non-SI metric unit of area equal to a square with 100-metre sides (1 hm), that is, 10,000 square metres (10,000 m), and is primarily used in the measurement of land. There are 100 hectares in one square kilometre. An acre is about 0.405 hectares and thus one hectare is about 2.47 acres.

In 1795, when the metric system was introduced, the are was defined as 100 square metres, or one square decametre, and the hectare ("hecto-" + "are") was thus 100 ares or ⁠1/100⁠ km (10000 square metres). When the metric system was further rationalised in 1960, resulting in the International System of Units (SI), the are was not included as a recognised unit. The hectare, however, remains as a non-SI unit accepted for use with the SI and whose use is "expected to continue indefinitely". Though the dekare/decare daa (1000 m) and are (100 m) are not officially "accepted for use", they are still used in some contexts.

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus. The former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus. They make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. It is the "mathematical backbone" for dealing with problems where variables change with time or another reference variable. It has also been called "the basic instrument of physical science".

The method of exhaustion (Latin: methodus exhaustionis) is a method of finding the area of a shape by inscribing inside it a sequence of polygons (one at a time) whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the nth polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes arbitrarily small, the possible values for the area of the shape are systematically "exhausted" by the lower bound areas successively established by the sequence members.

The method of exhaustion typically required a form of proof by contradiction, known as reductio ad absurdum. This amounts to finding an area of a region by first comparing it to the area of a second region, which can be "exhausted" so that its area becomes arbitrarily close to the true area. The proof involves assuming that the true area is greater than the second area, proving that assertion false, assuming it is less than the second area, then proving that assertion false, too.

In geometry, the area enclosed by a circle of radius r is πr. Here, the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.

One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons with an increasing number of sides. The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the circumference times the radius–namely, A = ⁠1/2⁠ × 2πr × r, holds for a circle.

The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration.

A general definition of surface area was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century. Their work led to the development of geometric measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface.

In geometry, a square is a regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal sides. As with all rectangles, a square's angles are right angles (90 degrees, or π/2 radians), making adjacent sides perpendicular. The area of a square is the side length multiplied by itself, and so in algebra, multiplying a number by itself is called squaring.

Equal squares can tile the plane edge-to-edge in the square tiling. Square tilings are ubiquitous in tiled floors and walls, graph paper, image pixels, and game boards. Square shapes are also often seen in building floor plans, origami paper, food servings, in graphic design and heraldry, and in instant photos and fine art.

In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex or star. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line), if the edge length is fixed.

Following are two lists of terrestrial lakes that have surface areas of more than approximately 3,000 square kilometres (1,200 sq mi), ranked by area, excluding reservoirs and lagoons.

The area of some lakes can vary over time, either seasonally or from year to year. This is especially true of salt lakes in arid climates.This list therefore excludes seasonal lakes such as Kati Thanda–Lake Eyre (maximum area 9,500 km, 3,700 sq mi), Mar Chiquita Lake (Córdoba) (maximum area 6,000 km, 2,300 sq mi), Chott el Djerid (up to 7,000 km, 2,700 sq mi, Lake Torrens (maximum area 5,745 km, 2,218 sq mi) and Great Salt Lake (maximum area, 1988, 8,500 km, 3,300 sq mi).