Question

How do you derive the area formula for a parallelogram?

Answer 1

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`