# Lesson 16

Applying Volume and Surface Area

### Lesson Narrative

In this second lesson on applying surface area and volume to solve problems, students solve more complex real-word problems that require them to choose which of the two quantities is appropriate for solving the problem, or whether both are appropriate for different aspects of the problem. They use previous work on ratios and proportional relationships, thus consolidating their knowledge and skill in that area. When students bring together knowledge of different areas of mathematics to solve a complex problem, they are engaging in MP4.

### Learning Goals

Teacher Facing

• Apply reasoning about surface area and volume of prisms as well as proportional relationships to calculate how much the material to build something will cost, and explain (orally and in writing) the solution method.

### Student Facing

Let's explore things that are proportional to volume or surface area.

### Student Facing

• I can solve problems involving the volume and surface area of children’s play structures.

### Glossary Entries

• base (of a prism or pyramid)

The word base can also refer to a face of a polyhedron.

A prism has two identical bases that are parallel. A pyramid has one base.

A prism or pyramid is named for the shape of its base.

• cross section

A cross section is the new face you see when you slice through a three-dimensional figure.

For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section.

• prism

A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.

Here are some drawings of prisms.

• pyramid

A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.

Here are some drawings of pyramids.

• surface area

The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron, without any gaps or overlaps.

For example, if the faces of a cube each have an area of 9 cm2, then the surface area of the cube is $$6 \boldcdot 9$$, or 54 cm2.

• volume

Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps.

For example, the volume of this rectangular prism is 60 units3, because it is composed of 3 layers that are each 20 units3.