# A Geometric Representation of the Distributive Property

The distributive property of multiplication over addition states that for all real numbers $a$, $b$ and $c$, then

$a(b + c) = ab + ac$.

In this short post, we are going to see the visual representation or ‘visual proof’ of this property where it is represented as area. However, one limitation of this representation is it does not represent negative values for $a$, $b$ or $c$. This means that this is only good for positive numbers.

A Geometric Representation of  the Distributive Property

In the diagram below, two rectangles (red and blue) are placed adjacently. The have the same height $a$. The red rectangle has width $b$ while the blue rectangle  has width $c$

Clearly, the area of the red rectangle is $ab$ and the area of the blue rectangle is $ac$. Also, the combined area of the red rectangle and the blue rectangle is

$ab + ac$.

Notice that the green rectangle has the same size as the combined red and blue rectangles. Therefore, its width is $b + c$ and its height is $a$.  So its area, which is the product of its width and height is

$a(b + c)$.

Now, since the are of the green rectangle is the same as the  combined area of the red and blue rectangles

$a(b + c) = ab + ac$

As  you can see, this is the distributive property of multiplication over addition.