Lesson 16
Distinguishing Between Surface Area and Volume
Lesson Narrative
In this optional lesson, students distinguish among measures of one, two, and threedimensional attributes and take a closer look at the distinction between surface area and volume (building on students' work in earlier grades). Use this lesson to reinforce the idea that length is a onedimensional attribute of geometric figures, surface area is a twodimensional attribute, and volume is a threedimensional attribute.
By building polyhedra, drawing representations of them, and calculating both surface area and volume, students see that different threedimensional figures can have the same volume but different surface areas, and vice versa. This is analogous to the fact that twodimensional figures can have the same area but different perimeters, and vice versa. Students must attend to units of measure throughout the lesson.
Note: Students will need to bring in a personal collection of 10–50 small objects ahead of time for the first lesson of the next unit. Examples include rocks, seashells, trading cards, or coins.
Learning Goals
Teacher Facing
 Comprehend that surface area and volume are two different attributes of threedimensional objects and are measured in different units.
 Describe (orally and in writing) shapes built out of cubes, including observations about their surface area and volume.
 Determine the surface area and volume of shapes made out of cubes.
Student Facing
Let’s contrast surface area and volume.
Required Materials
Required Preparation
 Prepare solutions to the first question of 123 Dimensional Attributes activity on a large visual display.
 Prepare sets of 16 snap cubes and two sticky notes for each student.
Learning Targets
Student Facing
 I can explain how it is possible for two polyhedra to have the same surface area but different volumes, or to have different surface areas but the same volume.
 I know how one, two, and threedimensional measurements and units are different.
CCSS Standards
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.

face
Each flat side of a polyhedron is called a face. For example, a cube has 6 faces, and they are all squares.

net
A net is a twodimensional figure that can be folded to make a polyhedron.
Here is a net for a cube.

polyhedron
A polyhedron is a closed, threedimensional shape with flat sides. When we have more than one polyhedron, we call them polyhedra.
Here are some drawings of polyhedra.

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 cm^{2}, then the surface area of the cube is \(6 \boldcdot 9\), or 54 cm^{2}.

volume
Volume is the number of cubic units that fill a threedimensional region, without any gaps or overlaps.
For example, the volume of this rectangular prism is 60 units^{3}, because it is composed of 3 layers that are each 20 units^{3}.
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