Lesson 15

More Nets, More Surface Area

Let’s draw nets and find the surface area of polyhedra.

15.1: Notice and Wonder: Wrapping Paper

Kiran is wrapping this box of sports cards as a present for a friend.

A prism with base length labeled 4 inches and base width labeled 2.5 inches.

What do you notice? What do you wonder?

15.2: Building Prisms and Pyramids

Your teacher will give you a drawing of a polyhedron. You will draw its net and calculate its surface area. 

  1. What polyhedron do you have?

  2. Study your polyhedron. Then, draw its net on graph paper. Use the side length of a grid square as the unit.

  3. Label each polygon on the net with a name or number.

  4. Find the surface area of your polyhedron. Show your thinking in an organized manner so that it can be followed by others.

15.3: Comparing Boxes

Here are the nets of three cardboard boxes that are all rectangular prisms. The boxes will be packed with 1-centimeter cubes. All lengths are in centimeters.

Three nets labeled A--C.
  1. Compare the surface areas of the boxes. Which box will use the least cardboard? Show your reasoning.
  2. Now compare the volumes of these boxes in cubic centimeters. Which box will hold the most 1-centimeter cubes? Show your reasoning.


Figure C shows a net of a cube. Draw a different net of a cube. Draw another one.  And then another one. How many different nets can be drawn and assembled into a cube?

Summary

The surface area of a polyhedron is the sum of the areas of all of the faces. Because a net shows us all faces of a polyhedron at once, it can help us find the surface area. We can find the areas of all polygons in the net and add them.

A square pyramid with a base of side length 6 and triangles of height 5.
A net with a square of side length 6 surrounded by triangles of height 5.

A square pyramid has a square and four triangles for its faces. Its surface area is the sum of the areas of the square base and the four triangular faces:

\((6\boldcdot 6) + 4\boldcdot \left(\frac12 \boldcdot 5 \boldcdot 6\right) = 96\)

The surface area of this square pyramid is 96 square units.

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.