Area
The area K of a trapezoid is given by
where a and b are the lengths of the parallel sides, and h is the height (the perpendicular distance between these sides.) In 499 AD Aryabhata, a great mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, used this method in the Aryabhatiya (section 2.8). This yields as a special case the well-known formula for the area of a triangle, by considering a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point.
Therefore the area of a trapezoid is equal to the length of this midsegment multiplied by the height
From the formula for the height, it can be concluded that the area can be expressed in terms of the four sides as
When one of the parallel sides has shrunk to a point (say a = 0), this formula reduces to Heron's formula for the area of a triangle.
Another equivalent formula for the area, which more closely resembles Heron's formula, is
where is the semiperimeter of the trapezoid. (This formula is similar to Brahmagupta's formula, but it differs from it, in that a trapezoid might not be cyclic (inscribed in a circle). The formula is also a special case of Bretschneider's formula for a general quadrilateral).
From Bretschneider's formula, it follows that
The line that joins the midpoints of the parallel sides, bisects the area.
Read more about this topic: Trapezoid
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