This activity reviews the work that students did previously when they drew shapes with given conditions. Students draw as many different triangles as they can that could be the base of the triangular prism, given two side lengths and one angle measurement for the triangle.

In preparation for calculating surface area and volume in the next activity, students select one of their triangles and find its area. This will require them to draw and measure the height of the triangle. As needed, remind students that the height must be perpendicular to whichever side they are using as the base of their triangle. Also, prompt students to measure the height as precisely as possible, because it will influence the accuracy of their later calculations.

The goal is to show the four possible triangles and make sure each student has calculated the area of their triangle correctly because this will affect their calculations in the next activity.

Ask students to share triangles they drew so that everyone has an opportunity to see all four triangles. If any of the four triangles are not presented by students, demonstrate how to construct it. Ensure that the class agrees that 4 unique triangles have the given measurements. The third side length of the triangle could be 5.0 cm, 7.3 cm, 2.6 cm, or 9.7 cm.

Make sure that students have calculated the area of their selected triangle correctly, because this will affect their volume and surface area calculations in the next activity.

In this activity, students take the triangle they selected in the previous activity and use it as the base of their triangular prism. 

The goal is to make sure that students understand how the changes in the base triangle affect the prism’s surface area and volume. Select students to share their answers for the cross-sections. For cross-sections taken in these two ways, all triangular prisms should have the same shapes as answers, although the actual size of the cross-section will differ based on the size of the base triangle and the height of the prism.

The volume and surface areas will depend on the triangle they have chosen to use as their base.

Select students to share their methods for computing volume and surface area. The base area is important in the calculation of each, so students should use the values they computed in the previous activity.

Students combine their solid with a partner’s and examine the new solid’s properties.

The purpose of this discussion is to explore which attributes of the larger prisms were easiest to determine based on the original prisms and which were hardest.

Using two prisms that are identical, demonstrate putting them together against a matching side in various ways.

For each configuration, discuss with students:

• “How does the area of the new prism compare with the area of the original prisms?”
• “How does the height of the new prism compare with the height of the original prisms?”
• “How does the volume of the new prism compare with the volume of the original prisms?”
• “How does the surface area of the new prism compare with the surface area of the original prisms?”

In the first configuration,

• The area of the base is the same as in the original prism.
• The height is twice the height of the original prism.
• The volume is twice the volume of the original prism.
• The surface area is less than twice the surface area of the original prism (because of the sides that are put together).

In the second and third configurations,

• The area of the base is twice the original.
• The height is the same as the original.
• The volume is twice the original.
• The surface area is less than twice the original.

How could you put these two prisms together to make the largest surface area possible for the new prism? The smallest surface area possible?

If time allows, this activity can be extended to review relationships between angles as well.

• Take two of the prisms and put them together so that their 45 angles are adjacent.
• Ask students what type of angle is created and what the relationship between the two angles creating that angle must be. (a right angle, complementary angles)
• Take three identical prisms and put them together so that a different angle from each prism is adjacent. Ask students what type of angle is created. (a straight angle). Ask students to identify pairs of angles that are supplementary.