How do you derive the area formula for a parallelogram?

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##