A trapezium, also known as a trapezoid, is a quadrilateral in which a pair of sides are parallel, but the other pair of opposite sides are non-parallel. The area of a trapezium is computed with the following formula:
The parallel sides are called the bases of the trapezium. Let and be the lengths of these bases. The distance between the bases is called the height of the trapezium. Let be this height. Then this formula becomes:
Given a trapezium, let and be the lengths of the bases, and let be the height. Draw a segment parallel to the bases that is halfway between the bases. This divides the trapezium into two trapeziums, each with the same height of
Labeling the angles of these trapeziums:
Note the following congruences and identities due to the fact that the bases are parallel:
Now rotate the top trapezoid and place it adjacent to the bottom trapezoid in the following way:
Due to the aforementioned congruences and identities, this shape is a parallelogram. The length of its base is and its height is This parallelogram has the same area as the trapezoid, so the area of the trapezoid is
Consider a trapezium in which . , and they are separated by a distance of . Find the area of .
We know that the area of a trapezium =
Substituting the values, we get,
Find the area of a trapezium with parallel lines of and , and a height of .