How do you derive the area formula for a parallelogram?

1 Answer
Dec 24, 2015

enter image source here

The intuition is fairly simple. In the above picture, we need to show that the area of parallelogram ABEC has the same area as the rectangle ABFD, which is bh.

Although it seems obvious from the picture, we cannot make the claim immediately without some justification. However, that justification comes fairly quickly when we show that triangles ACD and BEF are congruent.

To show that, we will use SSS congruence (two triangles with all three sides being equal are congruent). note that we immediately have bar(AC) = bar(BE) as opposite sides of parallelograms are equal and bar(BF) = bar(AD) by construction. Finally, to show that bar(CD) = bar(EF), we first observe that bar(CE) = bar(AB) = bar(DF), and therefore

bar(CD) = bar(CE) - bar(DE) = bar(DF) - bar(DE) = bar(EF).

With that, we have ACD ~= BEF. Thus

"area"(ABEC)="area"(ABED) + "area"(ACD)

="area"(ABED) + "area"(BEF)

="area"(ABFE)

=bh