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History of area measurement


What occupied the geometers of five thousand years ago? the answer is Earth measurement. The area problem has been studied since the ancient Babylonian. The methods used to calculate areas range from decomposing the complicated polygons/polyhedron into simpler regions, for example with parallel lines /planes or using triangulations, up to calculus methods.




  A(abcd)=1/4(a +c)(b+d)   

where a,b,c,d  is the sides of a quadrilateral. Unfortunately, this gives the correct result only in the case of the rectangle.


 The visual justification method were employed by Egyptians to calculate the area.



The isosceles triangle can be divided by altitude into two right triangles, then join to form a rectangle of height equal to the altitude and base equal to one-half the base of the triangle (see Fig.4).


The formula will give us the area of a semicircle, if r is the radius, or a hemisphere if x be the diameter. The value p comes from the following formula:  


The area of a circle of diameter 9 is calculated as that of a square of side 8 (see Fig.5), so the value p is:




where a,b,c are the sides and s is one-half of the diameter, i.e. s=(a + b + c)/2




where  a, b, c, d are edges of lengths of   cyclical quadrilaterals (see Fig.5)semiperimeter  s= (a + b + c + d)/2. This formula is an amazing symmetric formula. If one side is zero length, say d = 0, then we have a triangle (which is always cyclic) and this formula reduces to Heron's one.

Generalizations of formula for quadrilaterals



where a and b are two non-adjacent angles (see Fig.7).        



where p and q are the lengths of the diagonals (see Fig.8).






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