Editing Area (section)

====Dissection, parallelograms, and triangles====
{{Main article|Triangle area|Parallelogram#Area formula}}
[[File:ParallelogramArea.svg|thumb|right|upright|A parallelogram can be cut up and re-arranged to form a rectangle.]]

Most other simple formulas for area follow from the method of [[dissection (geometry)|dissection]].
This involves cutting a shape into pieces, whose areas must [[addition|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:<ref name=AF/>
:{{bigmath|''A'' {{=}} ''bh''}} &nbsp;(parallelogram).
[[File:TriangleArea.svg|thumb|right|upright|A parallelogram split into two equal triangles]]
However, the same parallelogram can also be cut along a [[diagonal]] into two [[congruence (geometry)|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:<ref name=AF/>
:<math>A = \frac{1}{2}bh</math> &nbsp;(triangle).
Similar arguments can be used to find area formulas for the [[trapezoid]]<ref>{{citation|title=Problem Solving Through Recreational Mathematics|title-link=Problem Solving Through Recreational Mathematics|first1=Bonnie|last1=Averbach|author1-link=Bonnie Averbach|first2=Orin|last2=Chein|publisher=Dover|year=2012|isbn=978-0-486-13174-0|page=[https://books.google.com/books?id=Dz_CAgAAQBAJ&pg=PA306 306]}}</ref> as well as more complicated [[polygon]]s.<ref>{{citation|title=Calculus for Scientists and Engineers: An Analytical Approach|first=K. D.|last=Joshi|publisher=CRC Press|year=2002|isbn=978-0-8493-1319-6|page=43|url=https://books.google.com/books?id=5SDcLHkelq4C&pg=PA43|url-status=live|archive-url=https://web.archive.org/web/20160505011253/https://books.google.com/books?id=5SDcLHkelq4C&pg=PA43|archive-date=2016-05-05}}</ref>