Question

Explain how the formula for the area of a trapezoid is derived from the formula for the area of a triangle?

Answer 1

Trapezoid can be divided into two triangles by a diagonal. These triangles will have bases that correspond to trapezoid's bases and altitudes equal to trapezoid's altitude.

Explanation:

One of the way to explain a formula for an area of a trapezoid using a formula for a triangle can be as follows.

Assume a trapezoid `ABCD` with lower base `AB` and upper base `CD` (they are parallel) and sides `AD` and `BC`.

Connect vertices `A` and `C` with a diagonal.
Consider triangle `Delta ABC` as having a base `AB` and an altitude from vertex `C` down to point `M` on base `AB` (`CM_|_AB`).
Its area is
`S_1 = 1/2*AB*CM`

Consider triangle `Delta BCD` as having a base `CD` and an altitude from vertex `B` up to point `N` on base `CD` (`BN_|_CD`).
Its area is
`S_2 = 1/2*CD*BN`

Altitudes `CM` and `BN` are equal and constitute the distance between two parallel bases `AB` and `CD`.
They both are equal to the altitude of the trapezoid `h`.

Therefore, we can represent areas of our two triangles as
`S_1 = 1/2*AB*h`
`S_2 = 1/2*CD*h`
Adding them together, we get the area of the whole trapezoid:
`S = S_1 + S_2 = 1/2(AB+CD)h`,
which is usually represented in words as "half-sum of the bases times the altitude".