A universal volume formula?

 

The formula

                    

                     For most regular solids standing upright, you can use the following universal formula for finding their volumes:

 

               

    where   h               is the height of the solid,

                ATOP           is the cross sectional area of the top of the solid,

                AMIDDLE      is the cross sectional area of the middle of the solid,

                ABOTTOM    is the cross sectional area of the bottom of the solid.

 

 

Sphere

                     h             =     2r,         

                     ATOP        =      0

                     AMIDDLE =    pr2

                     ABOTTOM =      0

     \ Volume of Sphere

    

   

   

 

 

r is the radius of the sphere

 

Cone

                     h             =     height of cone,    

                     ATOP        =      0

                     AMIDDLE =    p(r/2)2

                     ABOTTOM =      pr2

     \ Volume of Cone

    

   

   

 

 

r is the radius of the base of the cone

 

Conical frustum

                     h             =     height of frustum,       

                     ATOP        =      pa2

                     AMIDDLE =    p[(a+b)/2]2

                     ABOTTOM =      pb2

     \ Volume of Conical Frustum

    

                 Üsimplify yourselves!

   

 

 

 

a is the top radius

b is the base radius

 

Square frustum

                     h             =     height of frustum,       

                     ATOP        =      a2

                     AMIDDLE =    [(a+b)/2]2

                     ABOTTOM =      b2

     \ Volume of Square Frustum

    

        Üsimplify yourselves!

   

 

 

Harder exercise

 

             Can you verify the formulas for the volumes of Conical Frustum and Square Frustum by other method?