WOLFRAM|DEMONSTRATIONS PROJECT

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.
remove rectangles
remove brown triangles
remove cyan triangles
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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.