Lesson Volume of prisms

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Volume of prisms


Prisms are solid bodies with flat faces that have two congruent parallel faces and a set of parallel edges that connect corresponding vertices of the two parallel faces.
Figures  1a  -  1e  present the examples of prisms.




    

            Figure 1a. A cube          






  Figure 1b. A rectangular prism      



Figure 1c. A triangular prism      



Figure 1d. A quadrilatelar prism  



  Figure 1e. A hexagonal prism

In the school geometry,  only  upright prisms  are considered.  They are prisms that have each lateral face plane perpendicular to the plane  (to the planes)  of the bases.  It means,  in particular,  that each lateral face of an upright prism is a rectangle.  It also means that each lateral edge of the upright prism,  i.e. an intersection of two adjacent lateral faces,  is perpendicular to the planes of the bases.  All lateral edges of an upright prism are congruent.  Any of these edges is  (is called)  the  height of a prism.

The bases of a prism are parts of their planes restricted by polygons.  Depending on the shape of that polygons,  the prisms can be called  triangular prisms,  or  rectangular prisms,  or  pentagonal,  hexagonal  and so on.

This lesson is focused on calculating the volume of prisms.

Major formulas for calculating the volume of prisms


1.  Volume  V  of a cube  with the edge measure  a  is  V = a%5E3.

2.  Volume  V  of a rectangular prism  with the edge measures  u,  v  and  w  is  V = uvw%29.

3.  Volume  V  of an upright prism  is  V = S%2Ah,  where  S  is the area of the prism base and  h  is the prism height.


  (1)

  (2)

  (3)


All the formulas  (1),  (2)  and  (3)  are the direct consequences of the base postulates for volume of the lesson  What is volume?  under the topic  Volume, Metric volume  of the section  Geometry  in this site.


Example 1

Find the volume of a cube with the edge measure of  5 cm.

Solution

The volume of a cube with the side measure  a is a%5E3  cubic units.                                                

So,  in our case the volume of the cube is  V = 5%5E3 = 125 cm%5E3.

Answer.  The volume of the cube is  V = 125 cm%5E3.



Figure 2. To the Example 1


Example 2

Find the volume of a rectangular prism which has the edge measures of  8 cm,  10 cm  and  6 cm.

Solution

The volume of a rectangular prism with the side measures  u,  v and  w is                                  
u%2Av%2Aw  cubic units.

So,  in our case the volume of the rectangular prism is  V = 8%2A10%2A6 = 480 cm%5E3.

Answer.  The volume of the rectangular prism is  V = 480 cm%5E3.



    Figure 3. To the  Example 2


Example 3

Find the volume of a triangular prism if its base is an equilateral triangle with the side measure of  4 cm  and the height of the prism is of  10 cm.

Solution

First,  the area of the triangle at the base is
S%5Bbase%5D = 1%2F2.4.4sqrt%283%29%2F2 = 4%2Asqrt%283%29 = 4%2A1.732 = 6.928 cm%5E2 (approximately).                                              


Now,  the volume of the prism is  V = S%5Bbase%5D%2Ah = 6.928*10 =
                                                                        = 69.28 cm%5E3 (approximately).

Answer.  The volume of the prism is  69.28 cm%5E3 (approximately).



    Figure 4. To the  Example 3


Example 4

Find the volume of a hexagonal prism if its base is a regular hexagon with the side measure of  4 cm  and the height of the prism is of  10 cm.

Solution

First,  the area of the regular hexagon at the base is
S%5Bbase%5D = 6.(1%2F2.4.4sqrt%283%29%2F2) = 24sqrt%283%29 = 24%2A1.732 = 41.569 cm%5E2 (approximately).

Now,  the volume of the prism is  V = S%5Bbase%5D%2Ah = 41.569*10 =
                                                                          = 415.69 cm%5E3 (approximately).                              

Answer.  The volume of the prism is  415.69 cm%5E2 (approximately).



    Figure 5. To the  Example 4


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

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

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