נוסחאות יפות בגאומטריה

Abstract

As it is well known that the Heron formula is  used as an efficient tool for calculating areas of triangles with given sides. we can also easily notice that all   area formulas of  quadrilaterals depend, in general, in kind of quadrilateral, whether it is a parallelogram, trapezoid or kite etc. Here we ask: suppose that given all sides of a general quadrilateral , is it possible to calculate the area of that quadrilateral?

It is not difficult to reveal that knowing the lengths of the sides is not sufficient to calculate the area, because simply more than one quadrilateral can be built of four  given sides. Therefore, we can not talk about a general area formula for a quadrilateral when only given the lengths of the sides . In such quadrilateral it is easy to notice that its area changes according to its angles. i.e quadrilateral area depends on its angles .

Suppose given all measures of  quadrilateral sides and angles. what is the area formula of such quadrilateral?. That is what we try reveal in this paper. In fact , we will show that knowing the side lengths and the measures of  only two opposite angles are sufficient to build the area formula.

We will also deal with some  special cases and consequences that could be derived from the formula such as Heron's formula and other ones that we will show later.