"The Surface Area of a Cylinder" (Version 2.0)

Edwina R. Justice              Gunsaulus Scholastic Academy
                               4420 South Sacramento Ave.
                               Chicago IL   60632
                               (312) 535-7215

Introductory Comments:

This is a description of a phenomenological approach presentation.  It is an 
extension of a mini-teach, "The Area of a Circle" (1986) and a PA, "The Surface Area 
of a Cylinder" (1988).  A problem-solving situation, which requires the use of the 
two concepts, was formulated. 

Objectives:

1) Use a phenomenological approach to problem-solving.
2) Apply concepts to problem-solving situation.
3) Participate in group activity.

Materials:

Measuring tapes
Tape (masking or scotch)
Round container lids (different circumferences)
Paper circles (equal circumferences)
Construction paper
Rectangles with different dimensions (measure of base should be equal to measure of 
    circumference of a corresponding lid) 

Worksheet:
    Cut two circles with equal diameters and one rectangle from cm. grid paper.  
Measure of base of rectangle should correspond to measure of circumference of one 
circle.  These figures should be used to make a worksheet which can be distributed to 
each student. 

Recommended Strategy:

Form small groups and measure circumference and diameter of several lids. 

Divide circumference by diameter for each lid.

Discuss constant (pi) that results when circumference is divided by diameter. 

Use paper circles to show A = pi(r2). (This procedure is explained in SMILE, 1986.) 

Calculate areas of figures on worksheet.

Cut figures and make a cylinder.

Relate areas of plane figures to surface area of resulting cylinder. 

Use rectangles and lids to make several other cylinders.

Calculate the surface areas of the cylinders (Use area of each rectangle plus two 
   times area of corresponding lid). 

Discuss related equations:

   Area of rectangle section of cylinder = base * height  
                                               or
                                 circumference of lid * height

                           Circumference = pi(d)
                                         = 2(pi)r

                                Diameter = 2r

                                Diameter = c/pi


Develop formula for surface area of cylinder:

                         Area = (2(pi)r2) + (2(pi)rh)

Review use of 2 as a constant in equation:

      2(pi)r2   (two represents two circles)
      2(pi)rh    (two represents two radii or one diameter) 

Solve problem:

        Two rectangular sheets 20 cm. by 24 cm. and 15 cm. by 30 cm. are to be rolled 
to form cylinders.  What is the height and diameter of the cylinder with maximum 
surface area that can be formed using either of these sheets? 

Construct the cylinder with maximum surface area (with lids).

Report results to class.