Area of a Parallelogram

slide triangle

rotate (a, c)

This geometric Demonstration establishes that the area of a parallelogram bounded by vectors

(a,c)

and

(b,d)

is

|ad-bc|

.

Use the sliders to see how various parallelograms can be transformed into ones of equal area with their bases on the

x

axis. If the

x

axis does not intersect the parallelogram, slide the triangular portion farthest from the

x

axis toward it. If the axis does intersect the parallelogram, slide the triangular portion cut off by the

x

axis to the farther end of the parallelogram.

In the examples shown, the height of the resulting parallelogram is

|d|

and its base is determined by the

x

axis intersection of the line joining

(a,c)

to

(a+b,c+d)

, which is easily seen to be

(ad-bc)/d

. Thus the area is

|ad-bc|

.

Clearly the same result holds if, due to the choice of triangular portion, the roles of (

a

,

c

) and (

b

,

d

) are interchanged.