**6690 - Using Computers in Mathematics Instruction**

**Danie Brink **

**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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