Ashley,

I remember the area of a trapezoid as

the average of the lengths of the parallel sides times the distance between the parallel sides.

I can show you why this is true for one type of trapezoid and leave it to you to check for other trapezoids.

Suppose I have a trapezoid where both base angles are acute. Let s₁ and s₂ be the lengths of the parallel sides with s₁ ≥ s₂and d be the distance between the parallel sides.

Construct a rectangle with base s₁ and height d. The area of the trapezoid is the area of this rectangle minus the areas of the two triangles shaded orange in the diagram.

That is

trapezoid area = s₁ × d - 1/2 d × x - 1/2 d × y
= d × (s₁ - 1/2(x + y))

But s₁ - s₂ = x + y so

trapezoid area = d × (s₁ - 1/2(s₁ - s₂))
= 1/2 (s₁ + s₂) × d.

Penny