Area of Trapezium
Jump to navigation
Jump to search
Theorem
Let $ABCD$ be a trapezium:
and
- whose height is $h$.
Then the area of $ABCD$ is given by:
- $\Box ABCD = \dfrac {h \paren {a + b} } 2$
Proof
Extend line $AB$ to $E$ by length $a$.
Extend line $DC$ to $F$ by length $b$.
Then $BEFC$ is another trapezium whose parallel sides are of lengths $a$ and $b$ and whose height is $h$.
Also, $AEFD$ is a parallelogram which comprises the two trapezia $ABCD$ and $BEFC$.
So $\Box ABCD + \Box BEFC = \Box AEFD$ and $\Box ABCD = \Box BEFC$.
$AEFD$ is of altitude $h$ with sides of length $a + b$.
Thus from Area of Parallelogram the area of $AEFD$ is given by:
- $\Box AEFD = h \paren {a + b}$
It follows that $\Box ABCD = \dfrac {h \paren {a + b} } 2$
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: $4.7$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): trapezium (US: trapezoid)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): trapezium (US: trapezoid)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): trapezium (trapezia)
- 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