Editing Area (section)

===Area of curved shapes===

====Circles====
[[File:CircleArea.svg|thumb|right|alt=A circle divided into many sectors can be re-arranged roughly to form a parallelogram|A circle can be divided into [[Circular sector|sectors]] which rearrange to form an approximate [[parallelogram]].]]
{{main article|Area of a circle}}
The formula for the area of a [[circle]] (more properly called the area enclosed by a circle or the area of a [[disk (mathematics)|disk]]) is based on a similar method.  Given a circle of radius {{math|''r''}}, it is possible to partition the circle into [[Circular sector|sectors]], as shown in the figure to the right.  Each sector is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram.  The height of this parallelogram is {{math|''r''}}, and the width is half the [[circumference]] of the circle, or {{math|π''r''}}.  Thus, the total area of the circle is {{math|π''r''<sup>2</sup>}}:<ref name=AF/>
:{{bigmath|''A'' {{=}} π''r''<sup>2</sup>}} &nbsp;(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 (mathematics)|limit]] of the areas of the approximate parallelograms is exactly {{math|π''r''<sup>2</sup>}}, which is the area of the circle.<ref name=Surveyor/>

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]]:
:<math>A \;=\;2\int_{-r}^r \sqrt{r^2 - x^2}\,dx \;=\; \pi r^2.</math>

====Ellipses====
{{main article|Ellipse#Area}}
The formula for the area enclosed by an [[ellipse]] is related to the formula of a circle; for an ellipse with [[semi-major axis|semi-major]] and [[semi-minor axis|semi-minor]] axes {{math|''x''}} and {{math|''y''}} the formula is:<ref name=AF/>
:<math>A = \pi xy .</math>