Half-side formula

In <a href="/facts/Spherical_trigonometry/3G3Fhk2h">spherical trigonometry</a>, the half side formula relates the angles and lengths of the sides of <a href="/facts/Spherical_triangle/3G3Fhk2h">spherical triangles</a>, which are triangles drawn on the <a href="/facts/Spherical_geometry/WGLQb1Ys">surface of a sphere</a> and so have curved sides and do not obey the formulas for <a href="/facts/Triangle/cRXUwsMn">plane triangles</a>.
For a triangle 
 
 
 
 △
 A
 B
 C
 
 
 {\displaystyle \triangle ABC}
 
 on a sphere, the half-side formula is

tan
                ⁡
                
                  
                    
                      1
                      2
                    
                  
                
                a
              
              
                
                =
                
                  
                    
                      
                        −
                        cos
                        ⁡
                        (
                        S
                        )
                        
                        cos
                        ⁡
                        (
                        S
                        −
                        A
                        )
                      
                      
                        cos
                        ⁡
                        (
                        S
                        −
                        B
                        )
                        
                        cos
                        ⁡
                        (
                        S
                        −
                        C
                        )
                      
                    
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\tan {\tfrac {1}{2}}a&={\sqrt {\frac {-\cos(S)\,\cos(S-A)}{\cos(S-B)\,\cos(S-C)}}}\end{aligned}}}

where a, b, c are the angular lengths (measure of <a href="/facts/Central_angle/h00sYLNj">central angle</a>, arc lengths normalized to a <a href="/facts/Unit_sphere/5UeoTcug">sphere of unit radius</a>) of the sides opposite angles A, B, C respectively, and 
 
 
 
 S
 =
 
 
 
 1
 2
 
 
 
 (
 A
 +
 B
 +
 C
 )
 
 
 {\displaystyle S={\tfrac {1}{2}}(A+B+C)}
 
 is half the sum of the angles. Two more formulas can be obtained for 
 
 
 
 b
 
 
 {\displaystyle b}
 
 and 
 
 
 
 c
 
 
 {\displaystyle c}
 
 by permuting the labels 
 
 
 
 A
 ,
 B
 ,
 C
 .
 
 
 {\displaystyle A,B,C.}

The polar dual relationship for a spherical triangle is the half-angle formula, 

 
 
 
 
 
 
 
 tan
 ⁡
 
 
 
 1
 2
 
 
 
 A
 
 
 
 =
 
 
 
 
 sin
 ⁡
 (
 s
 −
 b
 )
 
 sin
 ⁡
 (
 s
 −
 c
 )
 
 
 sin
 ⁡
 (
 s
 )
 
 sin
 ⁡
 (
 s
 −
 a
 )
 
 
 
 
 
 
 
 
 
 
 {\displaystyle {\begin{aligned}\tan {\tfrac {1}{2}}A&={\sqrt {\frac {\sin(s-b)\,\sin(s-c)}{\sin(s)\,\sin(s-a)}}}\end{aligned}}}

where semiperimeter 
 
 
 
 s
 =
 
 
 
 1
 2
 
 
 
 (
 a
 +
 b
 +
 c
 )
 
 
 {\displaystyle s={\tfrac {1}{2}}(a+b+c)}
 
 is half the sum of the sides. Again, two more formulas can be obtained by permuting the labels 
 
 
 
 A
 ,
 B
 ,
 C
 .
 
 
 {\displaystyle A,B,C.}

Half-side formula open-in-new

Half-side formula