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Prisms, Pyramids and Cylinders

Thinking in Three Dimensions


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Copyright 2011 by Robert Leatherwood

The main difference between plane geometry and solid geometry is the figures of solid geometry occupy space.  Unlike figures considered in plane geometry, the solid figures are three-dimensional and as such, contain volume.  The volume is expressed in cubic units and represents the number of cubes having a unit-long edge that the solid can enclose.

The Prism

Prisms are polyhedrons that come in many different shapes and sizes.  They all have one thing in common – they have two bases.  The bases are parallel, congruent polygons and the shape of the base(s) is what gives the prism its name.  The other faces of the prism are called lateral faces. 

The altitude (height) of a prism is a perpendicular segment that joins the planes of the bases.

The lateral area of a prism is the sum of the areas of the lateral faces which is the sum of the height and perimeter.

The surface area is the sum of the lateral area and the area of the two bases.

The volume of a prism is equal to the product of a base and the height.

           

        Right Prism

The right prism is a three-dimensional figure in which the lateral edges form right angles with the bases.  This also means that, in the right prism, any lateral edge is also an altitude.

          Oblique Prism      

          The oblique prism differs from the right prism in that its lateral edges  are not perpendicular to the bases.  The altitude or height of the  oblique prism is a line segment that is perpendicular to, and defines  the distance between the bases.

                                  

                                   Right Prism                        Oblique Prism

Example    Find the total surface area and volume of a right triangular prism with sides of 5 cm, 12 cm, and 13 cm and a height of 14 cm.

                                                              

Since the bases are right triangles, we know that the hypotenuse is the longest side and therefore, the legs are 5 and 12 cm.

The area of each base is

 

The lateral area is the total lateral face area,  

    

The total surface area is  

The volume, .

 

The Pyramid

A pyramid is a polyhedron in which the base can be any polygon and the lateral faces are triangles that meet at the vertex of the pyramid.  The pyramid is named by the shape of its base.  The altitude of a pyramid is the perpendicular line segment from the vertex to the plane of the base and the length of the altitude is the height, h.

A regular pyramid is one whose base is a regular polygon and whose lateral faces are congruent isosceles triangles.  The slant height, , is the length of the altitude of a lateral face of the pyramid.

                    
        Regular Pyramid                                Oblique Pyramid

 

The lateral area of a pyramid is the sum of the areas of its lateral faces. For a regular pyramid, the area of each face is the triangular area given by the product of one-half the triangle’s base, s and its altitude (slant height),.  The total lateral area is the sum of the total number of faces having base s and altitude .  Since the total number of bases is actually the perimeter of the pyramid we can write,

                                        

The total surface area, SA is the sum of the lateral area and the base area:

                                     

The volume of the pyramid is: 

The foregoing formula for the volume works for all pyramids.

       

           Example    Given a regular square pyramid with side = 15 in, slant                                   height, = 9 in., and height, h = 7 in.

                Find the lateral area, surface area and volume.

               

The lateral area,

                The surface area,

                      The volume,

 

The Cylinder

The cylinder is like a prism in that it has two congruent parallel bases, the bases, however, are circular.  The altitude of a cylinder is a perpendicular segment that joins the planes of the bases and the height, h of the cylinder is the length of the altitude.  Note that the height or altitude is the length between the bases no matter the orientation of the cylinder.

 

                           Description: cylinders                                   
           Right Cylinder            Oblique Cylinder

 

If you were to “unroll” the curved surface of a cylinder you would find that it is a rectangle of height = h and a length equal to the perimeter of the base.

             

The area of that rectangle isthe lateral area of the cylinder:

                                          

As with the other polyhedrons we have discussed, the surface area of the cylinder is found by adding the two base areas to the lateral area:

                                           .

The volume of the cylinder is given by:  

 

       Example   Given a right cylinder with height = 8 in. and radius = 3 in.

Find the lateral surface area, the total surface area, and the volume of the cylinder to the nearest respectively.