12.1: Surface Area of a Cylinder
This activity is intended to supplement Geometry, Chapter 11, Lesson 2.
ID: 10075
Time required: 30 minutes
Activity Overview
Students will explore a net representation for a right cylinder. The surface area will be developed from the parts of the net.
Topic: 3Dimensional Geometry
 Construct 3 dimensional prisms and pyramids from nets.
 Calculate the surface area of a right prism or cylinder.
Teacher Preparation and Notes
 This activity is designed to be used in a high school or middle school geometry classroom.
 This activity is designed to be studentcentered.

The surface area of a right cylinder with base radius \begin{align*}= R\end{align*}
=R and height \begin{align*}= H\end{align*}=H is \begin{align*}SA = 2\pi R^2 + 2\pi RH\end{align*}SA=2πR2+2πRH . The activity asks students to notice that the circumference of the circle is the length of the rectangle in the net. 
The points \begin{align*}R\end{align*}
R and \begin{align*}H\end{align*}H control the radius and the height of the cylinder. When \begin{align*}R\end{align*}R is dragged, the length of the rectangle also changes (because the length = circumference of the circle). The height of the rectangle does not change when \begin{align*}R\end{align*}R is dragged.  Note: Measurements can display 0, 1, or 2 decimal digits. If 0 digits are displayed, the value shown will round from the actual value. To change the number of digits displayed:
 Move the cursor over the value so it is highlighted.
 Press + to display additional decimal digits or  to decrease digits.
 To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
 To download the calculator file, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=10075 and select CYLINDER.
Associated Materials
 Student Worksheet: Surface Area of Cylinders http://www.ck12.org/flexr/chapter/9696
 Cabri Jr. Application
 CYLINDER.8xv
Problem 1 – Nets
A net is a pattern that can be cut out and folded into a 3dimensional figure. Students should see a partial net of a right cylinder. If the rectangle of the net were rolled up, the circle would be the top face of the cylinder (like a lid of a jar).
The dimensions of the net can be changed by dragging the points \begin{align*}R\end{align*}
Students should use the D. & Length tool (\begin{align*}F5\end{align*}
Note: Display measurements with 2 decimal digits. To do this, hover the cursor over the measurement and then press the plus key (+).
Next, they should use the Calculate tool to divide the length of the rectangle by the radius of the circle to find that the width of the rectangle is the same as the circumference of the circle.
Problem 2 – Surface Area
Students should use the Area tool from the Measure menu (\begin{align*}F5\end{align*}
Note: Press the ALPHA button to access numerical characters. The tool icon in the corner of the screen will display \begin{align*} \begin{array} {c} ^1A\\
\hline \end{array}\end{align*}
Press the ENTER button to start and end the text.
Finally, students are to find the surface area of the cylinder. Remind students that this is only a partial net, so one of the faces is missing. To find the surface area, students need to find the area of the circle and the area of the rectangle. They should first use the Area tool from the Measurement menu to calculate these areas. Then, they need to use the Calculate tool to find the sum of both circle bases by clicking on the area of the circle, pressing \begin{align*}\downarrow\end{align*}