Download 6 Brahmagupta

July 2011
A
B
D
C
Quadrilateral ABCD is inscribed in the circle above.
How could you find the area of quadrilateral ABCD ?
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Brahmagupta (598–668) was an Indian mathematician
and astronomer. He developed a formula for finding
the area of any cyclic quadrilateral:
If you know the side lengths, then the area is
given by:
𝐴 = √(𝑡 − 𝑎)(𝑡 − 𝑏)(𝑡 − 𝑐)(𝑡 − 𝑑)
c
a
where t is the semi-perimeter.
i.e.
d
b
This formula does not work for most other quadrilaterals.
A cyclic quadrilateral is
a quadrilateral whose vertices
all lie on a single circle.
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Try Brahmagupta’s Formula
I.
Construct a cyclic quadrilateral using the TI-84’s Cabri Jr., the TI-Nspire,
Geometer’s Sketchpad or GeoGebra.
II.
Use the software to measure the side lengths and then apply
Brahmagupta’s Formula to find the area of the quadrilateral.
c
180 - 
a

e
d
b
III.
Verify the Area:
a.
On your constructed model, construct
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the diagonal e. Use the
software to measure the angle 𝛼 .
Why are opposite angles in a
cyclic quadrilateral supplementary?
Now use the triangle area formula
of the form
1
𝐴 = 𝑏𝑐 sin(𝛼)
2
to find the areas of the two triangles formed.
b.
On your constructed model, use the software to measure the
length e. Apply Heron’s formula twice to find the area of the
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quadrilateral.
c.
On your constructed model, declare the polygon to be a quadrilateral
and measure its area directly.
d.
How do the results of the four different area findings compare?
IV.
What connection do you see between Brahmagupta’s Formula and Heron’s
Formula?
V.
Develop a proof for Brahmagupta’s Formula.
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VI.
Use the Internet to Investigate Bretschneider's Formula which is an
extension of Brahmagupta’s Formula and works for finding the area of any
quadrilateral.
(Bretschneider’s work appeared in 1842 and later proved more completely
by Coolidge in 1939.)
Prove Bretschneider’s Formula.
Bretschneider's Formula
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Bretschneider's formula is the following expression for the area of
a quadrilateral,
Here, p, q, r and s are the sides of the quadrilateral, T is half the perimeter,
and A and C are two opposite angles
B
A
p
q
s
r
C
D
Bretschneider's formula works on any quadrilateral regardless of whether it
is cyclic or not.
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