We want to know if the areas recorded in the codices are correct.
To do that it is necessary to recall various properties about irregular polygons.

We will start with triangles and then quadrilaterals. We will show how to construct irregular quadrilaterals and give a program to construct arbitrary irregular polygons.

• Triangles :

  The triangle shown below is the only one with sides
  a, b, c clockwise.
  Side a is the one the x axis.

  We could calculate its area using the traditional formula
  (1/2)base times height, but we don't know the height. (Could you calculate it?)

  Luckily, Hero of Alexandria found a formula for the area (in square units) of a triangle in terms only of the lengths of its sides:

  A=
  √  s(s-a)(s-b)(s-c)  
  

  where s is the semiperimeter, that is

  
  s=(a+b+c)/2.
  

  In the figure
  you can change the sidelengths using the sliders.

  Play and observe.

  
  Observe:

  • The shape and area of the triangle depend only on the lengths (and order) of the sides.
  • If one side is larger than the sum of the others, no triangle can be formed (and the area
    appears as NaN, Not A Number).
  • What are the circles and what happens to them when the triangle disappears?
    
    
    
    
    
    The red circle has radius a and centre at the origin.
    The green circle has radius c. The intersection determines the position of vertex B. When there is no intersection there is no triangle.
    

    
    

  • For the 5,4,3 triangle, what is the angle ABC? How do you know? Can you calculate the area of the triangle without Heron's formula?
    
    
    
    
    
    Angle ABC is 90 degrees. We know that from Pythagoras' Theorem: in a right-angled triangle the sum of the squares of the sides equals the square of the hypothenuse.
    Since 3^2+4^2=5^2 the triangle must be right-angled.

    
    

  • Why don't we use the other intersection between the circles?
    
    
    
    
    
    
    Becasue it has the correct sideslengths in the wrong order.
    
  
  
  
  • Quadrilaterals :

    Now lets see what happens with quadrilaterals.

    The lengths of the sides do NOT determine shape or area.

    In the figure below, vertices B and C can be dragged to change the shape without
    changing the sidelengths.

    The area can be calculated in terms os sidelengths and angles using
    Bretschneider's formula:

    A=
    √  (s-a)(s-b)(s-c)(s-d)-abcd cos²((φ+ψ)/2)  
    
    where s is again the semiperimeter and

    φ, ψ are opposite angles of the quadrilateral.

    Play and observe.
    

    
    Observe:

    • Sidelengths do NOT determine area. What information do we have to add?
      
      
      
      
      We can add the values of two opposite angles or the two diagonals.
    • As before, if one side is larger that the sum of the others, no quadrilateral can be formed
      (and the area appears as NaN, Not A Number).
    • Sidelengths DO determine the maximum possible area. What is its formula?
      
      Amax=
      √  (s-a)(s-b)(s-c)(s-d)  
      .
    • The shape that gives the maximum area is the one inscribable in a circle.
      The circle shown is the one determined by A,B,C. The figure is inscribable when the circle also contains D.
      What do you observe about the angles? (in the figure they are given in radians) Answer here.

    
    
    

    Click to see the construction of an irregular quadrilateral

    Build your own polygon