Area of Parallelogram/Rectangle

From ProofWiki
Jump to navigation Jump to search


The area of a rectangle equals the product of one of its bases and the associated altitude.


Let $ABCD$ be a rectangle.


Then construct the square with side length:

$\paren {AB + BI}$

where $BI = BC$, as shown in the figure above.

Note that $\square CDEF$ and $\square BCHI$ are squares.


$\square ABCD \cong \square CHGF$

Since congruent shapes have the same area:

$\paren {ABCD} = \paren {CHGF}$ (where $\paren {FXYZ}$ denotes the area of the plane figure $FXYZ$).

Let $AB = a$ and $BI = b$.

Then the area of the square $AIGE$ is equal to:

\(\displaystyle \paren {a + b}^2\) \(=\) \(\displaystyle a^2 + 2 \paren {ABCD} + b^2\)
\(\displaystyle \paren {a^2 + 2 a b + b^2}\) \(=\) \(\displaystyle a^2 + 2 \paren {ABCD} + b^2\)
\(\displaystyle a b\) \(=\) \(\displaystyle \paren {ABCD}\)