Deriving the Formula for the Area of a Parallelogram

x

1

2

y

1

0.8

x

2

1.2

y

2

1.5

Calculate the area of the greenparallelogram by removing piecesfrom the big rectangle.

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(

x

1

+

x

2

)(

y

1

+

y

2

)

=

x

1

y

1

+

x

2

y

1

+

x

1

y

2

+

x

2

y

2

The Demonstration shows a graphics derivation of formula for the area of the parallelogram given by points

{0,0}

,

{

x

1

,

y

1

}

,

{

x

1

+

x

2

,

y

1

+

y

2

}

, and

{

x

2

,

y

2

}

, namely

x

1

y

2

-

x

2

y

1

. The area of triangle

{0,0}

,

{

x

1

,

y

1

}

,

{

x

1

+

x

2

,

y

1

+

y

2

}

is

(

x

1

y

2

-

x

2

y

1

)/2

. Since the expression

x

1

y

2

-

x

2

y

1

could be negative, it gives the so-called signed area.