EMAT 6690

Ken Montgomery

Brahmagupta’s Formula

Given Quadrilateral
ABCD, draw
diagonal_{}(Figure
1) dividing the quadrilateral into the two triangles,_{}and_{}.

**Figure
1:** Quadrilateral ABCD

Let the areas of _{}and_{}be
given by_{}and_{},
respectively. Then the total area of the quadrilateral (_{})
is given in Equation 1.

**Equation
1:** _{}

Applying trigonometry to_{}and_{},
respectively, we obtain the relationships given in
Equations 2 and 3.

**Equation
2: _{}**

**Equation
3: _{}**

Substitution of Equations 2 and 3 into Equation 1, yields Equation 4.

**Equation
4: _{}**

We, then square both sides in Equation 5.

**Equation
5: _{}**

Squaring the binomial term on the right side yields Equation 6.

**Equation
6: _{}**

Factoring out _{}gives
Equation 7.

**Equation
7: _{}**

From the Law of Cosines, we obtain Equations 8 and 9.

**Equation
8: _{}**

**Equation
9: _{}**

Setting Equations 8 and 9 equal, via the transitive property yields Equation 10.

**Equation
10: _{}**

Rearranging Equation 10, we obtain Equation 11.

**Equation
11: _{}**

We then square both sides of Equation 11.

**Equation
12: _{}**

Squaring the binomial term on the left yields Equation 13.

**Equation
13: _{}**

Next, we multiply both sides by c to obtain Equation 14.

**Equation
14: _{}**

Distributing on the left hand side gives Equation 15.

**Equation
15: _{}**

Factoring_{}out
of the left side results in Equation 16.

**Equation
16: _{}**

Rearranging Equation 16, we have Equation 17.

**Equation
17: _{}**

We restate Equation 7, here as Equation 18.

**Equation
18: _{}**

Adding zero to both sides, we have Equation 19.

**Equation
19: _{}**

Factoring, we have Equation 19, we obtain Equation 20.

**Equation
20: _{}**

Factoring, again, we have Equation 21.

**Equation
21: _{}**

Simplifying, via a
trigonometric
identity (_{}),
we have Equation 22.

**Equation
22: _{}**

We make use of another
trigonometric identity, given in Equation 23.

**Equation
23: _{}**

Applying Equation 23 to Equation 22, we obtain Equation 24.

**Equation
24: _{}**

Rearranging and factoring, we obtain Equation 25

**Equation
25: _{}**

We next multiply out
the term, ** _{}**in
Equation 26.

**Equation
26: _{}**

Distributing and simplifying, we have Equation 27.

**Equation
27: _{}**

Substitution into Equation 25, yields Equation 28.

**Equation
28: _{}**

Distributing the 4 gives Equation 29.

** Equation
29: _{}**

Simplifying, we have Equation 30.

** Equation
30:**

_{}

To factor the left
side, we add and
subtract _{},
obtaining Equation 31.

**Equation
31:**

_{}

Working with the numerator of the first fraction, we rearrange terms, applying the commutative property of addition, to obtain Equation 32.

** Equation
32:**

_{}_{}

Rewriting the terms,_{}and_{},
we have Equation 34.

**Equation
34: **

_{}_{}

Rewriting _{},
we obtain Equation 35.

** Equation
35: **

_{}_{}

Adding zero, via canceling terms, we rewrite Equation 35 equivalently in Equation 36.

**Equation
36: _{}**

Equation 36, then factors into the two products presented in Equation 37.

**Equation
37: _{}**

The right-hand side of Equation 38, will then factor into the four products of Equation 38.

**Equation
38: _{}**

Substituting from Equation 38, into Equation 31, we obtain Equation 39.

**Equation
39: _{}**

Writing 16 as its prime factorization, we obtain Equation 40.

**Equation
40:**

_{}

Rewriting –a as a – 2a,
-b as b –
2b, -c as c – 2c and –d as d – 2d, yields Equation 41.

**Equation
41:**

_{}

Dividing the right-most term in each of the four factors by 2, results in Equation 42.

**Equation
42:**

_{}

We define semi-perimeter in Equation 43.

**Equation
43: _{}**

Substituting from the relation in Equation 43, into Equation 42, we obtain Equation 44.

** **

**Equation
44: _{}**

Factoring out *abcd*
in the
second fraction, we obtain Equation 45.

**Equation
45: _{}**

Applying the Cosine
half-angle
trigonometric identity, results in Equation 46.

**Equation
46: _{}**

Taking the square root of both sides, we obtain the generalized formula of Brahmagupta, which does not require that the quadrilateral be circumscribed. This result is also known as Bretschnieder’s formula (Equation 47).

**Equation
47: _{}**

Open BrahmaguptasFormula.gsp
compare the calculated measurements of area.

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