Brahmagupta's Formula


Kala Fischbein and Tammy Brooks

Brahmagupta's Formula

Prove: For a cyclic quadrilateral with sides of length a, b, c, and d, the area is given by

where s is the semiperimeter.


Draw chord AC. Extend AB and CD so they meet at point P.

Angle ADC and Angle ABC subtend the same chord AC from the two arcs of the circle. Therefore they are supplementary. Angle ADP is supplementary to Angle ADC. So

Triangle PBC and Triangle PDA are similar. The ratio of similarity is

Area of Triangle PDA = * Area of Triangle PBC

Area ABCD = Area of Triangle PBC - Area of Triangle PDA

Let A = Area of quadrilateral ABCD and T = Area of Triangle PBC.

A = T - T = T = T

Let PA = e and PD = f. Applying Heron's Formula, the area of triangle PBC is


(Note: We have used s at this point for the semiperimeter of the TRIANGLE. In what follows, we will substitute for s, e, and f in terms of a, b, c, and d. Eventually we will return to the use of s to represent the semiperimeter of the quadrilateral. )

1. First, we want a substitution for e in terms of a, b, c, and d.

2. Next, we want a substitution for f in terms of a, b, c, and d.

3. Now we will make some substitutions in the formulas for the triangle.

4. Now evaluate s(s - (e + a)

5. Likewise, we now evaluate s - (f + c) in terms of a, b, c, and d.

6. Now, we evaluate (s - b)

7. Now we are ready to evaluate the Area of the quadrilateral in terms of a, b, c, d.


where s is the semiperimeter of the cyclic quadrilateral

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