Area of Parallelogram/Parallelogram

Theorem

Let $ABCD$ be a parallelogram whose adjacent sides are of length $a$ and $b$ enclosing an angle $\theta$.

The area of $ABCD$ equals the product of one of its bases and the associated altitude:

$\ds \map \Area {ABCD}$

$=$

$\ds b h$

$\ds $

$=$

$\ds a b \sin \theta$

where:

$b$ is the side of $ABCD$ which has been chosen to be the base

$h$ is the altitude of $ABCD$ from $b$.

Let $ABCD$ be the parallelogram whose area is being sought.

Let $F$ be the foot of the altitude from $C$

Also construct the point $E$ such that $DE$ is the altitude from $D$ (see figure above).

Extend $AB$ to $F$.

Then:

$\ds AD$

$\cong$

$\ds BC$

$\ds \angle AED$

$\cong$

$\ds \angle BFC$

$\ds DE$

$\cong$

$\ds CF$

Thus:

$\triangle AED \cong \triangle BFC \implies \map \Area {AED} = \map \Area {BFC}$

So:

$\ds \map \Area {ABCD}$

$=$

$\ds EF \cdot FC$

$\ds $

$=$

$\ds AB \cdot DE$

$\ds $

$=$

$\ds b h$

$\ds $

$=$

$\ds a b \sin \theta$

Definition of Sine of Angle: $h = a \sin \theta$

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: Parallelogram of Altitude $h$ and Base $b$: $4.3$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 7$: Geometric Formulas: Parallelogram of Altitude $h$ and Base $b$: $7.3.$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): parallelogram
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