

Area of an inscribed (cyclic) quadrilateral. Brahmagupta's Formula
A formula for calculating the area of an inscribed, or cyclic quadrilateral
when you know the lengths (a,b,c,d) of the sides.
Try this Drag any orange dot. Note the formula changes to calculate the area.
Recall that an
inscribed (or 'cyclic') quadrilateral is one where the four
vertices all lie on a circle.
Using the formula below, you can calculate the area of the quadrilateral.
where a,b,c,d are the side lengths, and p is half the perimeter:
In the figure above, drag any vertex around the circle. Note how the semiperimeter (p) and the area are calculated.
'Crossed' polygons
In the figure above, if you drag a point past its neighbor the quadrilateral will become 'crossed' where one side crossed over another.
In such 'crossed' quadrilaterals the area formula no longer holds.
(Most properties of polygons are invalid when the polygon is crossed).
Similarity to Heron's Formula
Recall that
Heron's formula for the area of a triangle is
where p is half the perimeter, as here.
The two formulae are very similar. If you take the Brahmagupta's formula and set d (the length of the fourth side) to zero,
the quadrilateral becomes a triangle. In Brahmagupta's formula, the term (pd) becomes just p and the formulae are then same also.
In the figure above, if you are careful, you can drag the point D around to be on top of A making d zero, illustrating this similarity.
From this you can see that Heron's formula is just a special case of Brahmagupta's. Recall too that all triangles are cyclic.
That is, you can always draw a circle through the three vertices. See
Circumcircle of a triangle.
Things to try
In the figure above
 Click on 'Hide details'.
 Drag the vertices around to create a new (uncrossed) quadrilateral.
 Calculate the area of the quadrilateral using Brahmagupta's Formula.
 Estimate the area by counting the squares. Each is one square unit.
 Click "show details" to check your answer.
While you are here..
... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone.
However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site?
When we reach the goal I will remove all advertising from the site.
It only takes a minute and any amount would be greatly appreciated.
Thank you for considering it! – John Page
Become a patron of the site at patreon.com/mathopenref
Other polygon topics
General
Types of polygon
Area of various polygon types
Perimeter of various polygon types
Angles associated with polygons
Named polygons
(C) 2011 Copyright Math Open Reference. All rights reserved

