LOJ 1331 - Agent J

Summary

In this problem, you have to find following area of 3 circle if they ar enclosed with each other

Hint

If you connect 3 center circle by line, a Tringle is form. Determine the area of circle and then minus the area of 3 sector of each circle

• Prerequisites-1 : Area of a sector of a circle
• Prerequisites-2 : Area of tringle if 3 sides given
• Prerequisites-3 : Angle if 3 sides given

Solution

Approach

A method for calculating the area of a triangle when you know the lengths of all three sides.

Let a,b,c be the lengths of the sides of a triangle. The area is given by:

where,


In our case the sides are a+b,b+c,c+a (See the picture above)

Let a be the opposite length of angle A. b,c are the side length of Angle A, The angle of each vertex of the triangle can be found by the following formula

In our case,
Opposite of angele A opp_A = b+c
Opposite of angele B opp_B = c+a
Opposite of angele C opp_C = a+b

After that find the Area of a sector of a circle from the following equation and subtract it form the total area


In our case the area of green sector of R1 circle,
r = a ,
theta = angle A

• Time Complexity: O(1) per test case.
• Memory Complexity: O(1) per test case.

Code

C++

#include <bits/stdc++.h>
using namespace std ;

double area(double r , double angle)
{
    return 0.5*r*r*angle ;
}

double angle(double opp_A, double opp_B, double opp_C)
{
    return acos( ( (opp_B*opp_B)+(opp_C*opp_C)-(opp_A*opp_A) ) / (2*opp_B*opp_C) ) ;
}


void _main_main()
{
    double a,b,c ;
    cin >> a >> b >> c ;

    double s = (a+a+b+b+c+c)/2.0 ;
    double total_area =  sqrt(s*(s-a-b)*(s-b-c)*(s-c-a)) ;

    total_area-= area(a,angle(b+c,c+a,a+b)) ;
    total_area-= area(b,angle(c+a,a+b,b+c)) ;
    total_area-= area(c,angle(a+b,b+c,c+a)) ;

    cout << fixed << setprecision(10) << total_area << "\n" ;

}

int main ()
{
    int testCase = 1 ; cin >> testCase ;
    for (int i = 0; i < testCase; i++){
        cout << "Case " << i+1 << ": " ;
        _main_main() ;
    }

}

Happy Coding!

Written by: Moontasir Mahmood

Tutorial source