## Thursday, October 21, 2010

### M6M4 ab: Surface area formulae

M6M4. Students will determine the surface area of solid figures (right rectangular prisms and cylinders).
a. Find the surface area of right rectangular prisms and cylinders using manipulatives and constructing nets.
b. Compute the surface area of right rectangular prisms and cylinders using formulae.
Surface area is simply the sum of the area of all the faces of a solid. Thus, as long as we can calculate the area of each face, there is nothing really new involved in this standard. The only tricky part here is to figure out the shape of the lateral face of a cylinder. But, students learn about the nets of various solids including cylinders in Grade 4. Thus, the surface area of a cylinder is the sum of the area of the two bases (circles) of the cylinder, and the area of the lateral face which is the rectangle with the dimensions equal to the height of the cylinder and the circumference of the base. We can summarize it in a formula like this:

Surface Area = 2 x (Area of base) + (Circumference of the base) x Height

Although I don't think it is that critical that students know this formula or the formula for prisms, it may be useful to have students explore the surface area of (rectangular) prisms not just as the sum of the areas of the faces. In fact, the first indicator discusses the use of nets in determining the surface area. If the surface area is simply the sum of the areas of the faces, there is really no need to use a net. So, what might be the reason for using nets to calculate the surface area of (rectangular) prisms?

We know that there are many different nets for a prism. However, a common net of a prism has all the lateral faces forming a "train" of rectangles and the two bases on the opposite sides of this "train" like this one.

Instead of calculating the area of each of the faces, you can consider the "train" of the lateral faces as one big rectangle, like this:

The length (vertical side in the drawing above) is equal to the height of the prism. The width (horizontal side) is actually the perimeter of the base. Thus, we can calculate the sum of the areas of the lateral faces as (Perimeter of the base) x height, too. But, then, the calculation of the surface area of a prism can be summarized in this formula:

Surface Area = 2 x (Area of base) + (Perimeter of base) x Height.

Perhaps investigating the surface area of prisms from this perspective allows us to use the same formula for all prisms and cylinders. However, I still don't think it is that important for students to know the formula...