§ 4 Volume, surface area and side area of ​​three-dimensional figures  

Geometric center of gravity and moment of inertia calculation formula

 

1.      Calculation formulas for volume, surface area, lateral area, geometric center of gravity and moment of inertia of three-dimensional figures

 

graphics

Volume V , surface area S , side area M , geometric center of gravity G and moment of inertia *J

a is the edge length, d is the diagonal

 

a, b, h are the length , width , height , respectively , d is the diagonal

 

 

volume 

surface area

side area

diagonal

The center of gravity G is at the intersection of the diagonals 

 

volume 

surface area

side area

diagonal

The center of gravity G is at the intersection of the diagonals 

Moment of inertia

Take the center of the cuboid as the origin of the coordinates , and the coordinates

The axis is parallel to the three edges

 

 

 

  

( At that time , it was the case of a cube )

 

In the table, m is the mass of the object, and the objects are all homogeneous . For the calculation formula of the moment of inertia of general objects, see Chapter VI, § 3 , 5 .

 

 

 

 

 

 

 

 

 

 

 

 

 

graphics

Volume V , surface area S , side area M , geometric center of gravity G and moment of inertia J

a,b,c are side lengths , h is height

 

a is the length of the base , h is the height , and d is the diagonal

 

n is the number of edges , a is the base length , h is the height , g is the oblique height

 

volume 

surface area

side area

    where F is the area of ​​the base

center of gravity 

    ( P and Q are the center of gravity of the upper and lower bottoms , respectively )

Moment of inertia

  For a regular triangular prism ( a=b=c ), take G as the coordinate origin , and the z - axis is parallel to the edge

     

volume 

surface area

side area

diagonal

center of gravity 

    ( P and Q are the center of gravity of the upper and lower bottoms , respectively )

Moment of inertia

  Take G as the origin of the coordinates , and the z - axis is parallel to the edge

     

 

volume 

surface area

side area

  where F is the area of ​​the base , which is the area of ​​the triangle on one side

Center of gravity ( Q is the center of gravity of the bottom surface )     

 

 

 

 

 

 

 

 

 

 

graphics

Volume V , surface area S , side area M , geometric center of gravity G and moment of inertia J

a,b,c,p,q,r is the edge length

 

 

 

h is high

 

 

 

a', a are the lengths of the upper and lower bases , n is the number of edges , h is the height , and g is the oblique height

volume

center of gravity  

       ( P is the vertex , Q is the center of gravity of the base )

 

 

 

 

 

volume 

where are the areas of the upper and lower bases , respectively

center of gravity 

    ( P, Q are the center of gravity of the upper and lower bottoms , respectively )

 

 

 

 

 

 

 

volume 

surface area

side area

  where are the areas of the upper and lower bases , respectively

center of gravity 

       ( P and Q are the center of gravity of the upper and lower bottoms , respectively )

 

 

 

 

 

 

 

 

graphics

Volume V , surface area S , side area M , geometric center of gravity G and moment of inertia J

The two bases are rectangular , a', b', a, b are the lengths of the upper and lower bases , respectively , h is the height , which is the length of the truncated edge

 

 

The base is a rectangle , a and b are the side lengths , h is the height , and a' is the upper edge length

 

 

 

r is the radius

volume

     

center of gravity 

      ( P, Q are the center of gravity of the upper and lower bottoms , respectively )

 

 

 

 

 

 

 

 

 

volume 

center of gravity 

   ( P is the midpoint of the upper edge , Q is the center of gravity of the lower base )

 

 

 

 

 

 

 

 

volume 

surface area 

The center of gravity G coincides with the center O of the sphere   

Moment of inertia

  Take the center of the ball O as the origin of the coordinates

     

     

 

 

 

 

 

 

graphics

Volume V , surface area S , side area M , geometric center of gravity G and moment of inertia J

[ hemisphere ]

r is the radius , O is the center of the sphere

 

 

   

 

r is the radius of the sphere , a is the radius of the bottom circle of the arch , h is the height of the arch , and is the cone angle ( radian )

 

r is the radius of the sphere , a is the radius of the arch bottom circle , and h is the arch height

volume 

surface area

side area

center of gravity 

Moment of inertia

  Take the center of the ball O as the origin of the coordinates , and the z - axis coincides with GO

     

     

 

 

volume 

surface area

Side area ( cone surface part )

center of gravity 

Moment of inertia

  z -axis coincides with GO

    

       

 

 

 

 

volume

surface area

Lateral area ( spherical part ) 

center of gravity 

 

 

 

 

 

graphics

Volume V , surface area S , side area M , geometric center of gravity G and moment of inertia J

[ table ]

r is the radius of the ball , a is the radius of the upper and lower base circles , and h is the height

 

 

 

 

 

R is the center radius , D is the center diameter , r is the radius of the circular section , d is the diameter of the circular section

 

 

 

volume 

surface area

side area

      

center of gravity 

      

       ( Q is the center of the bottom circle )

 

 

 

 

volume 

surface area

The center of gravity G is on the center of the ring 

Moment of inertia

  Take the center of the ring as the coordinate origin , and the z - axis is perpendicular to the plane where the ring is located

  

  

 

 

 

 

 

 

 

graphics

Volume V , surface area S , side area M , geometric center of gravity G and moment of inertia J

[ cylinder ]

r is the base radius , h is the height

R is the outer radius , r is the inner radius , h is the height

r is the radius of the base circle , h, H are the minimum and maximum heights , respectively, is the truncated angle , D is the axis of the truncated ellipse

volume   

surface area 

side area 

center of gravity   

       ( P and Q are the center of the upper and lower bottom circles , respectively )

Moment of inertia

  Take the center of gravity G as the origin of the coordinates , and the z - axis is perpendicular to the bottom surface

    

    

 

 

volume   

surface area 

side area 

     where t is the wall thickness of the pipe , and is the average radius

center of gravity   

Moment of inertia

  Take the z -axis coincident with GQ

    

volume   

surface area 

         

side area 

truncated ellipse axis 

center of gravity   

       

        ( GQ is the distance from the center of gravity to the bottom , GK

         is the distance from the center of gravity to the axis )

 

 

 

 

 

 

graphics

Volume V , surface area S , side area M , geometric center of gravity G and moment of inertia J

 

h is the maximum height of the section , b is the arch height of the bottom surface , 2 a is the chord length of the bottom surface , r is the radius of the bottom surface , and is the central angle ( radian ) opposite to the arc

 

a,b,c are half axes

 

volume   

 

   

Lateral area ( cylindrical part )

 

 

 

 

 

 

 

 

 

volume   

The center of gravity G is on the center O of the ellipsoid   

Moment of inertia

  Take the center of the ellipsoid as the origin of the coordinates , and the z -axis coincides with the c -axis

   

 

 

 

 

 

 

 

 

 

 

 

graphics

Volume V , surface area S , side area M , geometric center of gravity G and moment of inertia J

r is the radius of the base circle , h is the height , and l is the busbar

 

 

 

 

 

 

 

 

r, R are the upper and lower base circle radius , h is the height , l is the busbar

 

 

 

 

The upper and lower bases are parallel , , are the areas of the upper and lower bases, respectively , is the mid-section area , and h is the height

 

volume   

surface area 

side area 

busbar   

center of gravity   

        ( Q is the base circle center , O is the cone vertex )

Moment of inertia

  Take the vertex of the cone as the origin of coordinates , and the z - axis coincides with GQ

      

      

 

 

volume   

surface area 

side area 

busbar   

Cone height ( the distance from the intersection of the busbars to the bottom circle )

       

center of gravity   

        ( P and Q are the center of the upper and lower bottom circles , respectively )

 

volume   

 

 

 

[ Note ]   Pyramid, circular table, ball table, cone, prism, cylinder, etc. are all special cases of quasi-prism

 

 

 

 

 

graphics

Volume V , surface area S , side area M , geometric center of gravity G and moment of inertia J

d is the diameter of the upper and lower bottom circles , D is the diameter of the middle section , and h is the height

 

 

When the busbar is an arc :

volume

 

   

 

When the busbar is a parabola :

volume 

 

   

center of gravity 

      ( P and Q are the center of the upper and lower bottom circles , respectively )

 

 

2.      Polyhedron

 

 

[ regular tetrahedron ]

[ regular octahedron ]

[ regular dodecahedron ]

[ Icosahedron ]

graphics

face number f

4

8

12

20

edge number k

6

12

30

30

Vertex number e

4

6

20

12

Volume V

surface area S

In the table, a is the edge length .

[ Eulerian formula ]    The number of faces of a polyhedron is f , the number of edges is k , and the number of vertices is e .                       

 

 

Original text