Area

Area is the measurement of the size of a region on a surface. It describes how much two-dimensional space a shape covers. You can think of area as the amount of paint needed to cover a surface with one coat. It can also be the amount of material required to make a model of a shape. While length measures a one-dimensional line, area is a two-dimensional concept. If you measure the boundary of a three-dimensional object, that is called surface area.

To measure area, mathematicians compare shapes to squares of a fixed size. In the International System of Units (SI), the standard unit is the square metre (m²). A square metre is the area of a square with sides exactly one metre long.

Every unit of length has a corresponding unit of area. For example, you can use square centimetres (cm²), square millimetres (mm²), or square kilometres (km²). In non-metric systems, you might use square feet (ft²) or square miles (mi²). When converting, the area factor is the square of the length factor. For instance, there are 10 millimetres in a centimetre, but there are 100 square millimetres in one square centimetre.

Calculating area often involves using specific formulas for different shapes. The most basic formula is for a rectangle, where area equals length multiplied by width.

A square is a special rectangle where all sides are equal, so its area is the side length squared. Other shapes can be understood through a process called dissection. This means cutting a shape into smaller pieces and rearranging them. For example, a parallelogram can be cut and moved to form a rectangle.

A triangle can be seen as half of a parallelogram, so its area is half the base times the height.

You can find the area of any polygon by dividing it into several triangles.

Curved shapes require more advanced mathematical tools. For a circle, the area is related to its radius. You can imagine a circle divided into many small sectors. If you rearrange these sectors, they form an approximate parallelogram.

For more complex shapes with curved boundaries, mathematicians use calculus. The development of integral calculus in the late 17th century was largely motivated by the need to find these areas. Calculus allows us to find the area under a curve or the area between two different graphs.

Humans have been studying area for thousands of years. In the 5th century BCE, Hippocrates of Chios showed that the area of a disk is proportional to the square of its diameter. Eudoxus of Cnidus also discovered that the area is proportional to the radius squared. The mathematician Archimedes used Euclidean geometry to show that a circle's area equals a specific right triangle. This triangle has a base equal to the circle's circumference and a height equal to the radius. Archimedes also used a doubling method with polygons to approximate the area of a circle.

As time passed, new formulas were discovered for more complex polygons. In the 7th century CE, Brahmagupta created a formula for cyclic quadrilaterals, which are four-sided shapes inside a circle. Later, in 1842, Carl Anton Bretschneider and Karl Georg Christian von Staudt independently found a formula for any quadrilateral. In the 17th century, René Descartes developed Cartesian coordinates. This allowed mathematician Carl Friedrich Gauss to create the surveyor's formula in the 19th century. This formula can find the area of any polygon if you know the locations of its vertices.

Area is a fundamental concept used across many scientific fields. In geometry and calculus, it is essential for understanding shapes and curves. In linear algebra, area is related to the definition of determinants. In differential geometry, area is a basic property of surfaces. In advanced mathematical analysis, the area of a plane subset is defined using the Lebesgue measure. In higher mathematics, area is often viewed as a special case of volume for two-dimensional regions. Even in nuclear physics, scientists use a tiny unit called a barn to describe the cross-sectional area of interaction at the atomic scale.