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 Skip Navigation LinksMath Help > Geometry > Polygons and Triangles > Cyclic Quadrilateral > Circumscribed Cyclic Q.

The circumradius of a cyclic quadrilateral with consecutive sides a, b, c, d is given by

R = (1/4)sqrt((ac+bd)(ad+bc)(ab+cd)/((s-a)(s-b)(s-c)(s-d))), where s=(a+b+c+d)/2, the semiperimeter of the quadrilateral.

To find the circumradius of a cyclic quadrilateral, first consider the triangle formed by the first three vertices of the quadrilateral.  Circumscribe a circle around this triangle.  The radius of this circle will be the same as the radius of the circle circumscribed around the cyclic quadrilateral.
Now reveal the fourth vertex, D, on the circle, and draw chords from D to each of the other points.  Now ABCD is the cyclic quadrilateral, and AC and BD are its diagonals.  The lengths of the sides are a, b, c, and d, and the lengths of the diagonals are p and q.

Now that I've labeled the figure this way, the circumradius of the quadrilateral is the circumradius of the triangle ABC,

R = abp/(4K), where K is the area of triangle ABC.  (proof)

R = cdp/(4L), where L is the area of triangle CDA.

Let A be the area of the quadrilateral.  A = K+L

R = (abp+cdp)/(4A) = p(ab+cd)/(4A)

Similarly, R = q(bc + ad)/(4A), so

R2 = pq(ab+cd)(bc + ad)/(16A2)

Ptolemy's Theorem says ac+bd = pq, so R2 = (ac+bd)(ab+cd)(bc + ad)/(16A2)

Brahmagupta's Theorem says the area, A, of the quadrilateral is sqrt((s-a)(s-b)(s-c)(s-d)), so 
R2 = (ac+bd)(ab+cd)(bc + ad)/(16(s-a)(s-b)(s-c)(s-d)), then by taking the square root of each side, the result follows.

 

Related pages in this website:

Circumscribed Triangle -- proves the formula for the circumradius of a triangle.

Summary of geometrical theorems

Ptolemy's Theorem

Brahmagupta's formula

Triangle Home Page


The webmaster and author of this Math Help site is Graeme McRae.