graphics 

Calculation formula     

[ distance between two points ]

on flat surface

in space

[ The dividing line segment is the dividing point coordinates of the fixed ratio ]

          ( a )

          ( b )

  M ( x , y , z ) is the division point of A ₁ A ₂ , the division ratio      

on flat surface 

in space 

λ > 0 is called inner division and outer division . When λ = 1 , M is the midpoint of A ₁ A _{2 :}

      , ,

       graphics 

                Calculation formula     

[ Area of ​​a triangle on a plane ]

  A ₁ , A ₂ , A ₃ form a counterclockwise loop

    S _△ =

    When S _△ =0 , the three points of A ₁ , A ₂ , and A ₃ are collinear

[ Area of ​​a polygon on a plane ]

A ₁ , A ₂ , … , An form a _{counterclockwise} loop

S =

When S = 0 , A ₁ , A ₂ , … , A _n n points are collinear

[ volume of a tetrahedron in space ]

  right -handed system

V = =

When V = 0 , the four points M , A ₁ , A ₂ , and A ₃ are coplanar

[ Angle of the dihedral ]

  M ₀ , M ₁ , A ₁ , A ₂ are four non-coplanar points in space, q is the dihedral angle A ₁ - M ₀ M ₁ - A ₂ (0 ￡ q ￡ 180 ° )

When M ₀ is the origin, the positive direction of the x -axis, and the positive direction of the y -axis, then

When M ₀ is the origin and the positive direction of the x -axis, then

       graphics 

                Calculation formula     

[ Barycentric coordinates of mass point system ]

The mass of each particle in space is m _i , and the coordinates are ( x _i , y _i , z _i ) i =1, 2, … , n

The coordinates of the center of gravity G are

where ∑ represents the sum of i from 1 to n