Area of Parallelogram/Rectangle

Theorem

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:

$\map \Area {AB + BI}$

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

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

Thus:

$\square ABCD \cong \square CHGF$

Since congruent shapes have the same area:

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

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

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

$\ds \paren {a + b}^2$

$=$

$\ds a^2 + 2 \map \Area {ABCD} + b^2$

$\ds \paren {a^2 + 2 a b + b^2}$

$=$

$\ds a^2 + 2 \map \Area {ABCD} + b^2$

$\ds a b$

$=$

$\ds \map \Area {ABCD}$

$\blacksquare$

1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text I$: $\S 1$. Area of a Circle
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: Rectangle of Length $b$ and Width $a$: $4.1$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): area
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): area
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 7$: Geometric Formulas: Rectangle of Length $b$ and Width $a$: $7.1.$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $1$: Areas and volumes
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $1$: Areas and volumes