The experiment above is very simple. Click 3 times on the blue ruler at the bottom of the screen to determine the three lengths of the sides of a triangle. Once you do that, the triangle will be assembled for you, and its area will be printed using the formula on the previous page.

 

      You may check that using the yellow command field. If you want to calculate something, type calc <expression> on one line and press Enter, where <expression> is what you want to calculate. Use an asterisk to denote multiplication, i.e., 2*2 = 4, so calc 2*2 causes 4 to be printed. Use the calculator to calculate the area the old fashioned way.

 

      Now, you are probably wondering where this magic formula came from. We are going to take a little detour to show that it has a natural setting, not in the calculation of areas of triangles, but in the calculation of areas of quadrilaterals!  That may seem surprising, so we will take a little time to discuss how one calculates the area of a quadrilateral.

 

       First of all, a quadrilateral is a four-sided figure.  If we call the sides:  a,b,c and d (and use these symbols also to represent their lengths) then we  see that the choice of four nonnegative values for those numbers may or  may not determine a quadrilateral.  There is a condition, similar to the  triangle inequality that must be satisfied.  It says, in essence, that the sum

of lengths of any three sides must exceed the length of the fourth.  We state the condition in terms of the semi-perimeter again.  Here goes. 

 

        The semi-perimeter determined by four nonnegative numbers: a,b,c, and d is    

 

 

Call the semi-perimeters. Then the condition says:

 

 

 

        We will always assume that this condition is met for our lengths.  Still  the values of the four lengths will not be enough to determine a unique quadrilateral.  There will in general be many quadrilaterals with sides of  those lengths.  Even if we arrange the lengths in a fixed order around the quadrilateral (in say, the counterclockwise sense), there will still be many such quadrilaterals.  And they will have different areas.  We can express this by saying that a quadrilateral is not rigid.  Its shape may be changed without changing the length of any of its sides.  For example, the parallelograms:

 

  and 

 

have the same side lengths,  but obviously have different shape and areas.

 

        The question that we shall ask and answer on this, and the next  few pages is:

 

         Question:  Given four lengths: a,b,c, and d that satisfy the condition

                            stated above,   what is the largest area of any quadrilateral

                            that can be constructed using those lengths.

       

         When we discover the answer to this maximization problem, we will see immediately why Heron's formula is true! 

 

          So let us see what is involved in calculating the area of a quadrilateral.   Suppose given the following quadrilateral:

 

 

            Here, we will denote sides with lowercase letters, and angles with uppercase letters.  We see that we may calculate the area by drawing the diagonal e, and then by calculating the areas of the two resulting triangles.

 

       In fact, the area of this quadrilateral is just

 

(2.1)

 

 

       This expresses the area in terms of a,b,c,d and the angles T and P.   Obviously we need more than the sides alone to determine the area in light  of our observations above.  The remarkable thing is that, if we seek the largest area obtainable, then that depends only on a,b,c, and d, as we shall see.  

 

       Now, once a,b,c, and d are given, then T and P are not arbitrary.  They are related to each other by an equation.  This is evident from the fact that once the value of, say T is determined, then the value of P is (essentially) determined also.  (Actually, it may generally have two values, but only one will be natural if the shape is to vary smoothly with T).  We derive the relation between T and P below:  

 

       There is a very handy formula called the Law of Cosines that allows us to calculate the third side of a triangle if the other two, and the angle between them are known.  For example, for  above, it says:

 

       Law of Cosines:         

 

And for ,  it implies:       

 

      Thus, the relation between T and P is this:

 

 

 

    We rewrite this as relation

 

(2.2)

 

 

 

      For fixed values of a,b,c,d,  (2.2) determines a curve in the (T,P) plane. Each value of T between  and  determines a value for P between  and  and vice-versa, and these codetermined values define points on this curve.  Our job then is to locate points on the curve for which the area (given by (2.1)) is maximized, and then to determine the area at those points.  In the next experiment, we will create quadrilaterals, see how the areas vary as we deform them, and generate points on the curve defined by (2.2).