Lesson Volume of cylinders

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Volume of cylinders


Cylinder  is a  3D  solid body which is bounded by two congruent circular bases in parallel planes and the lateral surface consisting of parallel straight segments that connect the corresponding points at the base circles. The  Figure 1a  represents an example of a cylinder.  You may think that the circle  O2  is obtained from the circle  O1  by the parallel transfer along the straight line  O1O2  connecting the centers of the circles.  Then the corresponding points at the base circles are those obtained by this transfer.

  
          Figure 1a. Cylinder
          (general definition)

      
          Figure 1b. Right circular
                        cylinder

In the school geometry,  only the  right circular cylinders  are considered.  They are cylinders that have the axis  (the straight line connecting the centers of the circles)  perpendicular to the plane  (to the planes)  of the bases  (Figure 1b).  It means,  as a consequence,  that each lateral generatrix of a right circular cylinder is perpendicular to the planes of the bases.  All lateral generatrices of an right circular cylinder are congruent.  Any of these generatrices is  (is called)  the  height of a cylinder.

You may think a right circular cylinder as a  3D  solid body which is swept by a rectangle rotating in  3D  space around some of its sides as an sxis when the rectangle makes the full revolution (the full turn).

This lesson is focused on calculating the volume of the right circular cylinders.


Formula for calculating the volume of cylinders


The volume of a cylinder  is  V =  pir%5E2h = S%5Bbase%5Dh,
where  r  is the radius of the cylinder,  h  is its height,  and  S%5Bbase%5D= pir%5E2  is the base area  (the area of the circle at the base).


Example 1

Find the volume of a cylinder if its radius is of  5 cm  and the height is of  4 cm.

Solution

The volume of the cylinder is

V = pir%5E2h = 3.14159*5%5E2*4 = 3.14159*25*4 = 3.14159*100 = 314.159 cm%5E2 (approximately).

Answer.  The volume of the cylinder is  314.159 cm%5E2 (approximately).


Example 2

Two cylinders are joined in a way that the base of one cylinder is overposed on the base of the other as shown in the  Figure 2a.
The radius of one cylinder is  5 cm  and the height is  2 cm.  The radius of the other cylinder is  2 cm  and the height is  5 cm.
Find the volume of the composite body.

Solution

The  Figure 2b  represents the side view of the two cylinders.                  
The common axis is shown in blue in the Figures  2a  and  2b.

The volume under consideration is composed of the volume of the first                                
cylinder and the volume of the second cylinder:

V = pi%2Ar%5B1%5D%5E2h%5B1%5D + pi%2Ar%5B2%5D%5E2%2Ah%5B2%5D = pi.5%5E2.2 + pi%2A2%5E2.5 = pi.( 25%2A2+%2B+4%2A5 ) =
= 70pi = 70*3.14159 = 219.91 cm%5E2.

Answer.  The volume of the composite body is  219.91 cm%5E2 (approximately).



Figure 2a.  To the  Example 2      




      Figure 2b.  Side view
      of the two cylinders

Note.  The assumption that the cylinders are co-axial is not necessary.  The result is valid for non-axial cylinders too.


Example 3

Find the volume of the solid body concluded between two co-axial cylindrical surfaces  (Figure 3)  of the radii of  10 cm  and  5 cm  respectively if the common height of the two cylindrical shells is of  8 cm.

Solution

The  Figure 3  represents the solid body concluded between two co-axial cylindrical                      
surfaces.

Their common axis is shown in blue in this  Figure.

The volume under consideration is the volume of the larger cylinder minus the volume
of the smaller one,  i.e.

V = pi%2A10%5E2%2A8 - pi%2A5%5E2%2A8 = pi%2A%28100+-+25%29%2A8 = 3.14159*75*8 = 1884.95 cm%5E3 (approximately).

Answer.  The volume of the solid body under consideration is  1884.95 cm%5E3 (approximately).



Figure 3.  To the  Example 3


Example 4

Four through cylindrical holes are made in the solid cylinder parallel to its axis of symmetry  (Figure 4).
Find the volume of the obtained solid body if the diameter of the original cylinder is  10 cm,  its height is  8 cm  and the diameter of each hole is  2 cm.

Solution

The volume of the original solid cylinder is  V = pir%5E2h = pi*5%5E2*8 = 200%2Api.                                  

The volume of each of four holes is  V%5Bhole%5D = pi*1%5E2*8 = 8%2Api.

Hence,  the volume of the solid body under consideration is

V - 4%2AV%5Bhole%5D = 200%2Api - 4%2A8%2Api = 168%2Api = 527.52 cm%5E3 (approximately).

Answer.  The volume of the solid body under consideration is  527.52 cm%5E3 (approximately).



Figure 4.  To the  Example 4


Example 5

A through cylindrical hole is made in a rectangular prism  (rectangular box)  of dimensions  6x8x10 cm  along its axis of symmetry parallel to the shortest edge  (Figure 5).
Find the volume of the obtained solid body if the diameter of the hole is  2 cm.

Solution

The volume of the original rectangular prism is  V%5Bprism%5D = 6.8.10 = 480cm%5E3.                                  

The volume of the cylindrical hole is  V%5Bhole%5D = pir%5E2h = pi*1%5E2*6 = 6%2Api.

Hence,  the volume of the solid body under consideration is

V = V%5Bprism%5D - V%5Bhole%5D = 480 - 6%2Api = 480 - 6%2A3.14 = 461.16 cm%5E3 (approximately).

Answer.  The volume of the solid body under consideration is  461.16 cm%5E3 (approximately).



    Figure 3. To the  Example 4


Example 6

A pie,  which has a cylindrical shape,  is cut in  8  equal sectorial pieces along the radii.  Find the volume of each piece if the diameter of the original pie is of  10 inches  and its height is of  2 inches.

Solution

The volume of each piece is  1%2F8  of the volume of the entire pie,  which is the cylinder
of the diameter of  10 inches  and the height of  2 inches.  The volume of the entire pie is

V = pir%5E2h = pi*5%5E2*2 = 50pi  cubic inches,

and the volume of each piece is

1%2F8V = %2850%2A3.14%29%2F8 = 19.625 cubic inches.

Answer.  The volume of each piece of the pie is  19.625 cubic inches.


My lessons on volume of cylinders and other 3D solid bodies in this site are

Lessons on volume of prisms

Volume of prisms
Solved problems on volume of prisms
Overview of lessons on volume of prisms                    

Lessons on volume of pyramids

Volume of pyramids
Solved problems on volume of pyramids
Overview of lessons on volume of pyramids

Lessons on volume of cylinders

Volume of cylinders
Solved problems on volume of cylinders
Overview of lessons on volume of cylinders                

Lessons on volume of cones

Volume of cones
Solved problems on volume of cones
Overview of lessons on volume of cones                    

Lessons on volume of spheres

Volume of spheres
Solved problems on volume of spheres
Overview of lessons on volume of spheres


To navigate over all topics/lessons of the Online Geometry Textbook use this file/link  GEOMETRY - YOUR ONLINE TEXTBOOK.

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