# Spoj TETRA solution. Spoj Sphere in a tetrahedron solution.

Category: AdHoc, Geometry, Formula

This question is based on geometry as we need to find the radius of the sphere subscribed inside an irregular tetrahedron

*Required Radius = (3*volume of Tetrahedron)/(sum of surface areas of triangular faces).*

This requires calculation of volume of Irregular Tetrahedron using its edge lengths.

This question can be seen as the extension of Spoj Pyramids Problem, in which volume of tetrahedron is calculated.

### Spoj TETRA solution code:

#include <bits/stdc++.h> using namespace std; /*Heron's Formula to find area of a triangluar face*/ double areaHeron(double a1,double a2,double a3) { double s=(a1+a2+a3)/2.0; return sqrt(s*(s-a1)*(s-a2)*(s-a3)); } int main() { std::ios::sync_with_stdio(false); int t; cin>>t; while(t--) { double u,v,w,U,V,W,vol,a,b=12,total_area=0; cin>>u>>v>>w>>W>>V>>U; /*Adding total area of all sides*/ total_area += areaHeron(u,V,w); total_area += areaHeron(W,u,v); total_area += areaHeron(W,V,U); total_area += areaHeron(U,v,w); /*steps to calculate volume of a Tetrahedron using formula*/ a=4*(pow(u,2)*pow(v,2)*pow(w,2)) - pow(u,2)*pow((pow(v,2)+pow(w,2)-pow(U,2)),2) - pow(v,2)*pow((pow(w,2)+pow(u,2)-pow(V,2)),2) - pow(w,2)*pow((pow(u,2)+pow(v,2)-pow(W,2)),2) + ( pow(v,2)+pow(w,2)-pow(U,2))* (pow(w,2)+pow(u,2)-pow(V,2))* (pow(u,2)+pow(v,2)-pow(W,2) ); vol = sqrt(a); vol /= b; /*the radius of the inscribed circle is (3*volume)/total_area */ cout << std::fixed << std::setprecision(4) << vol*3/total_area<<"\n"; } return 0; }