Lesson 16

Distinguishing Between Surface Area and Volume

Lesson Narrative

In this optional lesson, students distinguish among measures of one-, two-, and three-dimensional 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 one-dimensional attribute of geometric figures, surface area is a two-dimensional attribute, and volume is a three-dimensional attribute.

By building polyhedra, drawing representations of them, and calculating both surface area and volume, students see that different three-dimensional figures can have the same volume but different surface areas, and vice versa. This is analogous to the fact that two-dimensional 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 three-dimensional 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 Preparation

  • Prepare solutions to the first question of 1-2-3 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 three-dimensional 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.

    Two figures, a pentagonal prism and a hexagonal pyramid.
  • 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 two-dimensional figure that can be folded to make a polyhedron.

    Here is a net for a cube.

    Six squares arranged with 4 in a row, 1 above the second square in the row, and one below the second square in the row.
  • polyhedron

    A polyhedron is a closed, three-dimensional shape with flat sides. When we have more than one polyhedron, we call them polyhedra.

    Here are some drawings of polyhedra.

    3 polyhedra, from left to right shapes resemble a house, drum, and star.
  • 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.

    A triangular prism, a pentagonal prism, and a rectangular prism.
  • 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.

    a rectangular pyramid, a hexagonal pyramid, a heptagonal pyramid
  • 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.

    Two images. First, a prism made of cubes stacked 5 wide, 4 deep, 3 tall. Second, each of the layers of the prism is separated to show 3 prisms 5 wide, 4 deep, 1 tall.