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# 12.1: Surface Area of a Cylinder

Difficulty Level: At Grade Created by: CK-12

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: 3-Dimensional 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 student-centered.
• The surface area of a right cylinder with base radius =R\begin{align*}= R\end{align*} and height =H\begin{align*}= H\end{align*} is SA=2πR2+2πRH\begin{align*}SA = 2\pi R^2 + 2\pi RH\end{align*}. The activity asks students to notice that the circumference of the circle is the length of the rectangle in the net.
• The points R\begin{align*}R\end{align*} and H\begin{align*}H\end{align*} control the radius and the height of the cylinder. When R\begin{align*}R\end{align*} 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 R\begin{align*}R\end{align*} 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:
1. Move the cursor over the value so it is highlighted.
2. Press + to display additional decimal digits or - to decrease digits.

Associated Materials

## Problem 1 – Nets

A net is a pattern that can be cut out and folded into a 3-dimensional 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 R\begin{align*}R\end{align*} and H\begin{align*}H\end{align*}. Students should drag these points and notice what changes with the figure for each point. When R\begin{align*}R\end{align*} is moved, the radius of the circle and the width of the rectangle are changed.

Students should use the D. & Length tool (F5\begin{align*}F5\end{align*} > Measure) to find the length of the rectangle.

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 (F5\begin{align*}F5\end{align*} > Measure) to find the areas of the rectangle and the circle. Then, they should use the Alph-Num tool to place the number 2 on the screen.

Note: Press the ALPHA button to access numerical characters. The tool icon in the corner of the screen will display 1A\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*}, and clicking on the 2. Next, they should click on the number they just calculated, press +, and click on the area of the rectangle.

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