A very long name

The area A of cyclic quadrilateral with sides a,b,c and d is given by $A=\sqrt { (s-a)(s-b)(s-c)(s-d) }$ where $s=\frac { (a+b+c+d) }{ 2 }$ Now, substituting the value of s in A , we get $A=\frac { 1 }{ 4 } \sqrt { (a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d) } \\ =\frac { 1 }{ 4 } \sqrt { 16\times 30\times 35\times 42 } \\ =\frac { 840 }{ 4 } =210$

PROOF of the formula:

Assume A,B,C and D to be the vertices of the cyclic quadrilateral and p,q,r,s be the lengths of sides AD,AB,BC and CD . $Area\quad of\quad ABCD=Area\quad of\quad \Delta ADB+Area\quad of\quad \Delta BDC\\ =\frac { pqsinA }{ 2 } +\frac { rssinC }{ 2 }$

Since, the quadrilateral is cyclic, so SinA=SinC $A=\frac { pqsinA }{ 2 } +\frac { rssinA }{ 2 } \\ { A }^{ 2 }=\frac { 1 }{ 4 } { \left( pq+rs \right) }^{ 2 }\sin ^{ 2 }{ A } \\ 4A^{ 2 }={ \left( pq+rs \right) }^{ 2 }\left( 1-\cos ^{ 2 }{ A } \right) \\ 4A^{ 2 }={ \left( pq+rs \right) }^{ 2 }-{ \left( pq+rs \right) }^{ 2 }\left( \cos ^{ 2 }{ A } \right)$

Solving for common side DB , in triangle ADB & BDC , using laws of cosines , we get ${ p }^{ 2 }+{ q }^{ 2 }-2pqcosA={ r }^{ 2 }+{ s }^{ 2 }-2rscosC$

As angles A and C are supplementary , cosA=-cosC and $2(pq+rs)cosA={ p }^{ 2 }+{ q }^{ 2 }-{ r }^{ 2 }-{ s }^{ 2 }$

Substituting this in the equation of area, we get $16{ A }^{ 2 }={ 4(pq+rs) }^{ 2 }-{ ({ p }^{ 2 }+{ q }^{ 2 }-{ r }^{ 2 }-{ s }^{ 2 }) }^{ 2 }$

The R.H.S. of the equation is of the form ${ a }^{ 2 }-{ b }^{ 2 }$ , so is can be written as $(2(pq+rs)-({ p }^{ 2 }+{ q }^{ 2 }-{ r }^{ 2 }-{ s }^{ 2 }))(2(pq+rs)+({ p }^{ 2 }+{ q }^{ 2 }-{ r }^{ 2 }-{ s }^{ 2 }))\\ =\left[ { \left( r+s \right) }^{ 2 }-{ \left( p-q \right) }^{ 2 } \right] \left[ { \left( p+q \right) }^{ 2 }-{ \left( r-s \right) }^{ 2 } \right] \\ =(p+q+r-s)(p+q-r+s)(p-q+r+s)(-p+q+r+s)$

So, the area of cyclic quadrilateral with sides a,b,c and d can be written as $A=\frac { 1 }{ 4 } \sqrt { (a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d) }$

Use Brahmgupta's Formula for finding the area of a cyclic quadrilateral. 4 x Area^2 = (a + b + c - d) ( a + b - c + d) ( a - b + c + d) ( -a + b + c + d).