If diameter of the circle entered in a trapeze — only the known size, then a problem of finding of the area of a trapeze has a set of decisions. The result depends on the size of corners between the basis of a trapeze and its sides.
Instruction
1. If it is possible to enter a circle in a trapeze, then the sum of sides is equal in such trapeze to the sum of the bases. It is known that the area of a trapeze is equal to the work of the half-sum of the bases on height. It is obvious that diameter of the circle entered in a trapeze is height of this trapeze. Then the area of a trapeze is equal to the work of the half-sum of sides on diameter of an inscribed circle.
2. Diameter of a circle is equal to two radiuses, and the radius of an inscribed circle — size known. There are no other data in a statement of the problem.
3. Draw a square and enter in it a circle. It is obvious that diameter of an inscribed circle is equal to the party of a square. Now present that two opposite sides of a square suddenly lost stability and began to tend to a vertical axis of symmetry of a figure. Such swaying is possible only at increase in the size of the party of the quadrangle described around a circle.
4. If two remained parties of the former square kept parallelism, the quadrangle turned into a trapeze. The circle becomes entered in a trapeze, diameter of a circle at the same time becomes height of this trapeze, and the parties of a trapeze got the different sizes.
5. Sides of a trapeze can creep away further. The point of contact will move on a circle. The parties of a trapeze in the swaying submit to only one equality: the sum of sides is equal to the sum of the bases.
6. It is possible to bring definiteness in the geometrical disorder formed by the unsteady parties if to know tilt angles of sides of a trapeze to the basis. Designate these corners α and β. Then after simple transformations the area of a trapeze can be written down the following formula: S=D (Sinα+Sinβ)/2SinαSinβгде S — the area of a trapeze of D — diameter of the circle entered in a trapeze α and β — corners between sides of a trapeze and its basis.