Area of a trapezoid. Reducing to the area of a rectangle

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Area of a trapezoid equals a product of the arithmetic mean of its bases' lengths and its height. Or, to put it short, ”half a sum of the bases times the height”.

This formula can be illustrated (or derived if it was forgotten) using the formula for the area of a rectangle. To do so, the trapezoid should be cut in such a way that the parts rearranged form a rectangle.

Drop the altitudes to the longer base at the midpoints of trapezoid's legs and then cut the trapezoid along the altitudes. Attach the two cut off right triangles's hypothenuses to the remaining parts of the legs. The shape obtained is a rectangle.

One of the rectangle's sides is as long as $h$ — the trapezoid's height. Sum of two other sides' lengths equals sum of the trapezoid's bases' lengths, thus one side is as long as half of that sum, that is $(a+b)/2$. So the triangle's area, and so the area of the initial trapezoid, equals $ S = (a+b) / 2 \cdot h$.

To prove that completely one still should make sure that the shape obtained after rearranging the triangles is indeed a rectangle — each short side and each composed long side is a line segment, and the corresponding lines are parallel. The fact that the angles obtained are right is provided by the cutting method itself — cuts are made along the altitude perpendicular to the base.

A model can be manufactured from an approximately 10 mm thick wooden board. It is convenient to attach the cut off triangles to the remaining part of the trapezoid with magnets: these should be put in the triangles' catheti, so that the original trapezoid can be obtained, as well as in the hypothenuses — so that the rectangle can be obtained.