# 11.2.1: Surface Area of a Cylinder

Section 11.2, "Surface Area of a Cylinder", is the second section of chapter 11, "Geometry and Measurement Relationships". The learning goal for this section is to "Develop and apply a formula for calculating the surface area of a cylinder", and as such, is where we really start to talk about cylinders in particular, rather than area and surface area in general. We start this section with an example where we are wrapping a couple small cylindrical objects (tea lights) into one big cylinder with wrapping paper. During this process, we realize that we can break down the surface area of a cylinder into the area of a rectangle and the area of the top and bottom circles.

Khan Academy has a fantastic breakdown of how this process works. Here are links to some of the more pertinent sections. Go directly to "Current Knowledge" for a direct explanation of the properties of cylinders.

**Prior Knowledge.** These lessons review what you learned in grade 7.

- Nets of Polyhedra. Examines the surface area of 3-dimensional shapes that have flat surfaces and straight edges.
- Nets of 3D Figures. Exercises that test your ability to visualize objects in 3D.
- Finding Surface Area: Nets of Polyhedra. Goes over an example that shows how to find surface area based on a net.
- Surface Area Using Nets. Exercises based on the previous video.
- Find the Volume of a Triangular Prism and Cube. This is a good warm-up video to calculating the surface area of a cylinder, since it uses the same process, but with a simpler object.

**Current Knowledge.** These lessons cover what you'll learn in grade 8.

- Cylinder Volume and Surface Area. Explains the properties of cylinders.
- Solid Geometry. These problems help verify your understanding of how to calculate surface area and volume.
- How Many People Can a Blimp Carry? Examines how density works in a real-world situation.
- Surface and Volume Density Word Problems. These questions help verify understanding of the previous video.
- Volume of a Cone. Explains the properties of cones.
- Volume of a Sphere. This video goes over an example of how to calculate the volume of a sphere.
- Volume Word Problems With Cones, Cylinders and Spheres. These problems help verify your understanding of the previous two videos.

**Calculating the Surface Area of a Cylinder**

The basic logic of calculating the surface area of a cylinder is this:

- The area of a circle is given by the formula, $latex A_{circle} = \pi r^2$
- The lateral surface of the cylinder, if viewed as a 2D surface, is a rectangle. Moreover, its base is the circumference of either circle. The area of the rectangle is given by the formula $latex A_{rectangle} = b \cdot h$, with $latex b = 2 \pi r$ (since the base of the rectangle is the circumference of the circle)
- Thus, the total surface area is $latex A_{total} = 2 \cdot A_{circle} + A_{rectangle}$ (the area of the top and bottom circles, but the area of the lateral rectangle)
- In other words, $latex A_{total} = 2 \pi r^2 + 2 \pi rh$

This, however, is a grade 10 way of looking at it. In grade 8, students simply need to be able to recognize that in order to find the area of the cylinder, they need to find the area of the top, the bottom, and the side, and add them all together.

**Support Questions**

Students can start working on these problems in the Nelson Mathematics 8 book. Please note that we will be evaluating student cahiers at the end of the chapter. If you want a level 4+ on your cahier evaluation, please make sure that you do (almost) all the support questions, and that you show your work and labels. The rule of thumb is this: you've shown your work properly if a grade 7 student can follow (though not necessarily understand) how you solved the problem. In other words, label your steps, show your formulas and (if necessary), draw a sketch.

- p. 376, #4 - 13