6690 - Using Computers in Mathematics Instruction

Danie Brink

 

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Essay 2:

Brahmagupta's Formula

Brahmagupta's formula finds the area of a cyclic quadrilateral. This links well with theorems 4, 5 and 6 of the instructional unit on this website. These theorems all deal with qualities of cyclic quadrilaterals. The formula for the area of a cyclic quadrilateral with sides a, b, c, d is given by

with s the semi-perimeter which is calculated by

Heron's Formula

There are various ways to find the area of a triangle. The method used to determine the area of a triangle largely depends upon the information that is known about the triangle. For example, if the length of the base of a triangle and the corresponding perpendicular height is known, we normally use the formula of half the length of the base multiplied by the perpendicular height of the triangle. When the length of the three sides are known, we can use the cosine rule and the area rule to determine the area of the triangle. Here are examples of these two cases. This is, however, a lengthy process and can be substituted by Heron's formula.

Heron's Formula calculates the area of a triangle with sides a, b, c by

with s the semi-perimeter which is calculated by

Some Interesting Facts

1. In essence the formulae of Bahmagupta and Heron are the same. When the side d in the cyclic quadrilateral collapses, the quadrilateral becomes a triangle. The two formulae are therefore exactly the same when we express Heron's formula as follows:

2. The right triangle with side lengths 3,4,5 has area 6 and is the only triangle with consecutive integer sides and area. The proof of this is straightforward:

Let a = n - 1, b = n, c = n + 1, Area = n + 2 substitute these into Heron's equation and find the positive integer root to the cubic . The result of interest is the only integer solution n = 4. You can click here to see the interesting graph of this cubic equation and confirm the integer solution of n = 4.

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Link to my 6690 Main Page