the area of a trapezoid
A trapezoid (called a trapezium in British usage) is a quadrilateral with one pair of parallel sides — think of a bucket seen from the side, wide at the top and narrow at the bottom, or a section of a canal. Its area lies between that of the two rectangles you could build on its short and long parallel sides.
Call the two parallel sides a and b, and call h the perpendicular distance between them (the height). The area is A = (1/2) · (a + b) · h. In words: average the two parallel sides, then multiply by the height. The averaging makes sense — the figure is 'between' a rectangle of width a and one of width b, so its effective width is their mean. One clean proof copies the trapezoid, rotates it 180 degrees, and fits the two together into a parallelogram of base (a + b) and height h, whose area is twice the trapezoid's.
If the two parallel sides happen to be equal, the trapezoid is really a parallelogram and the formula collapses to (1/2)(a + a)h = a · h, exactly as it should. The 'segment between' midline of a trapezoid — the segment joining the midpoints of the slanted sides — has length (a + b)/2, the very average that appears in the area.
A trapezoid with parallel sides 6 m and 10 m, and height 4 m, has area (1/2)(6 + 10)(4) = (1/2)(16)(4) = 32 m^2.
Average the two parallel sides, then multiply by the perpendicular height.
Only the parallel sides go into the average; the slanted (non-parallel) sides do not, and h must be the perpendicular distance between the parallel sides.