§ 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 .