Calculate area of a cyclic quadrilateral with given side lengths

Given four positive integers A, B, C, and D representing the length of sides of a Cyclic Quadrilateral, the task is to find the area of the Cyclic Quadrilateral.

Examples:

Input: A = 10, B = 15, C = 20, D = 25
Output: 273.861

Input: A = 10, B = 30, C = 50, D = 20
Output: 443.706

Approach: The given problem can be solved based on the following observations:

• A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. The circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic.

• In the above image above r is the radius of the circumcircle and A, B, C, and D are the lengths of the sides PQ, QR, RS, and SP respectively.
• The area of the quadrilateral is given by Bretschneider’s formula is:

where, A, B, C, and D are the sides of the triangle and
? and ? are the opposite angles of the quadrilateral.

Since, the sum of opposite angles of the quadrilateral is 180 degree. Therefore, the value of cos(180/2) = cos(90) = 0.

Therefore, the formula for finding the area reduces to .

Therefore, the idea is to print the value of  as the resultant area of the given quadrilateral.

Below is the implementation of the above approach:

273.861

Time Complexity: O(1)
Auxiliary Space: O(1)