Cyclic Quadrilateral Formulas

Posted In: Mathematics

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic.

Cyclic Quadrilateral


Sides of quadrilateral: a,b,c,d
Diagonals: d1,d2
Angle between the diagonals: φ
Internal angles: α,β,γ,δ
Radius of circumscribed circle: R
Perimeter: L
Semiperimeter: p
Area: S

  1. $α+β=γ+δ=180°$
  2. Ptolemy’s Theorem
    $ac+bd=d_1d_2$
  3. $L=a+b+c+d$
  4. $R=1/4√{{(ac+bd)(ad+bc)(ab+cd)}/{(p-a)(p-b)(p-c)(p-d)}}$
    where $p=L/2$
  5. $S=1/2d_1d_2\ sin\ φ$
    $S=√{(p-a)(p-b)(p-c)(p-d)}$
    where $p=L/2$

Post Tags: Geometry


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