Area of Parallelogram
Theorem
The area of a parallelogram equals the product of one of its bases and the associated altitude.
Proof
There are three cases to be analysed: the square, the rectangle and the general parallelogram.
Square
From Area of Square:
- $\paren {ABCD} = a^2$
where $a$ is the length of one of the sides of the square.
The altitude of a square is the same as its base.
Hence the result.
$\blacksquare$
Rectangle
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:
\(\ds \paren {a + b}^2\) | \(=\) | \(\ds a^2 + 2 \map \Area {ABCD} + b^2\) | ||||||||||||
\(\ds \paren {a^2 + 2 a b + b^2}\) | \(=\) | \(\ds a^2 + 2 \map \Area {ABCD} + b^2\) | ||||||||||||
\(\ds a b\) | \(=\) | \(\ds \map \Area {ABCD}\) |
$\blacksquare$
Parallelogram
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: $h = a \sin \theta$ |
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): parallelogram
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): parallelogram