Nets for Pyramids and Prisms

Task

1. Below is a net for a three dimensional shape:

   
   

   The inner quadrilateral is a square and the four triangles all have the same size and shape.

   1. What three dimensional shape does this net make? Explain.
   2. If the side length of the square is 2 units and the height of the triangles is 3 units, what is the surface area of this shape?

2. Draw a net for a rectangular prism whose base is a one inch by one inch square and whose faces are 3 inches by 1 inch.

   1. Is there more than one possible net for this shape? Explain.
   2. What is the surface area of the prism?

IM Commentary

The goal of this task is to work with nets for three-dimensional shapes and use them to calculate surface area. There are several extension problems available for this task including the following.

• How many different nets is it possible to make for the rectangular prism in part (b)? Students will need to decide what it means for two nets to be the same and will have to experiment to find all of the possibilities.
• With a single sheet of 8$\frac{1}{2}$ inch by 11 inch piece of paper, what is the largest possible rectangular prism that can be constructed? Here "largest possible" also needs to be interpreted as it could be understood to mean largest possible volume, largest possible surface area, or perhaps largest face.

These are challenging problems ideally suited for group work, probably over several class periods.

Solution

1. 1. This pattern will make a square pyramid, that is a pyramid with a square base. The four triangular sides all meet at the apex of the pyramid.
   2. The area of the square base will be 4 square units. The four faces of the pyramid all have the same area. Since each base is 2 units and each height is 3 units the  area of one triangle is $\frac{1}{2} \times 2 \times 3 = 3$ square units. So all four triangular faces have an area of 12 square units, and the total surface area of the pyramid is 4 + 12 = 16 square units.

2. An example of a net is pictured here:

   
       

   1. There are many different nets for this rectangular prism. We could, for example, leave the four rectangles one on top of the other as in the picture above and move the two square bases: the only restraint is that they need to share a side with one of the four rectangles and they cannot both be on the left or both be on the right. There are other possible nets as well, such as the one pictured below:

      
          

      Each square is one inch by one inch and each rectangle is 3 inches by 1 inch.

   2. To calculate the surface area of this rectangular prism we add the areas of the square bases (1 square inch each) to the areas of the rectangular faces (3 square inches each). There are two square bases and 4 rectangular faces so the total area is 2 $\times$ 1 + 4 $\times$ 3 = 14 square inches.