Area is the size that a plane surface or two-dimensional shape takes up between its boundary lines (length and breadth). Area is calculated by multiplying the length with the breadth. The blocks in the figure below shows
The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (m2), which is the area of a square whose sides are one metre long.[1] A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.
There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.
For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.
Every unit of length has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measure in square metres (m2), square centimetres (cm2), square millimetres (mm2), square kilometres (km2), square feet (ft2), square yards (yd2), square miles (mi2), and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units.
The SI unit of area is the square metre, which is considered an SI derived unit.
The conversion between two square units is the square of the conversion between the corresponding length units.
Example,
Since 1 foot = 12 inches,
The relationship between square feet and square inches is
1 square foot = 144 square inches,
where 144 = 122 = 12 × 12. Similarly:
- 1 square kilometre = 1,000,000 square meters
- 1 square meter = 10,000 square centimetres = 1,000,000 square millimetres
- 1 square centimetre = 100 square millimetres
- 1 square yard = 9 square feet
- 1 square mile = 3,097,600 square yards = 27,878,400 square feet
In addition,
- 1 square inch = 6.4516 square centimetres
- 1 square foot = 0.09290304 square metres
- 1 square yard = 0.83612736 square metres
- 1 square mile = 2.589988110336 square kilometres
There are several other common units for area. 1 are = 100 square metres
Though they are fallen out of use, the hectare is still commonly used to measure land:
- 1 hectare = 100 acres = 10,000 square metres = 0.01 square kilometres
Other uncommon metric units of area include the tetrad, the hectad, and the myriad.
The acre is also commonly used to measure land areas, where
- 1 acre = 4,840 square yards = 43,560 square feet.
An acre is approximately 40% of a hectare.
On the atomic scale, area is measured in units of barns, such that,
- 1 barn = 10−28 square meters.
The barn is commonly used in describing the cross sectional area of interaction in nuclear physics.
Basic area formula
The area of this rectangle is length x width.
The most basic area formula is the formula for the area of a rectangle. Given a rectangle with length l and w, the formula for the area is
A = lw (rectangle).
That is, the area of the rectangle is the length multiplied by the width. As a special case, the area of a square with side length s is given by the formula
A = s2 (square).
The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom. On the other hand, if geometry is developed before arithmetic, this formula can be used to define multiplication of real numbers.
Most other simple formulae for area follow from the method of dissection. This involves cutting a shape into pieces, whose areas must sum to the area of the original shape.
For an example, any parallelogram can be subdivided into a trapezoid and a right triangle, as shown in figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle:
A = bh (parallelogram).
Two equal triangles
However, the same parallelogram can also be cut along a diagonal into two congruent triangles, as shown in the figure to the right. It follows that the area of each triangle is half the area of the parallelogram:
(triangle)
Similar arguments can be used to find area formulae for the trapezoid and the rhombus, as well as more complicated polygons.
A circle can be divided into sectors which rearrange to form an approximate parallelogram.
Main article: Area of a circle
The formula for the area of a circle is based on a similar method. Given a circle of radius r, it is possible to partition the circle into sectors, as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form and approximate parallelogram. The height of this parallelogram is r, and the width is half the circumference of the circle, or πr. Thus, the total area of the circle is r × πr, or πr2:
A = πr2 (circle).
Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit of the areas of the approximate parallelograms is exactly πr2, which is the area of the circle.
This argument is actually a simple application of the ideas of calculus. In ancient times, the method of exhaustion was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to integral calculus. Using modern methods, the area of a circle can be computed using a definite integral:
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