# 1.5.4: More Nets, More Surface Area

- Page ID
- 39644

## Lesson

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

Exercise \(\PageIndex{1}\): Notice and Wonder: Wrapping Paper

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

What do you notice? What do you wonder?

Exercise \(\PageIndex{2}\): Building Prisms and Pyramids

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

- What polyhedron do you have?
- Study your polyhedron. Then, draw its net on graph paper. Use the side length of a grid square as the unit.
- Label each polygon on the net with a name or number.
- Find the surface area of your polyhedron. Show your thinking in an organized manner so that it can be followed by others.

Exercise \(\PageIndex{3}\): 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.

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

Are you ready for more?

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 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:

\((6\cdot 6)+4\cdot (\frac{1}{2}\cdot 5\cdot 6)=96\)

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

### Glossary Entries

Definition: 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.

Definition: Face

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

Definition: Net

A net is a two-dimensional figure that can be folded to make a polyhedron.

Here is a net for a cube.

Definition: 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.

Definition: 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.

Definition: 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.

Definition: 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 \(6\cdot 9\), or 54 cm^{2}.

## Practice

Exercise \(\PageIndex{4}\)

Jada drew a net for a polyhedron and calculated its surface area.

- What polyhedron can be assembled from this net?
- Jada made some mistakes in her area calculation. What were the mistakes?
- Find the surface area of the polyhedron. Show your reasoning.

Exercise \(\PageIndex{5}\)

A cereal box is 8 inches by 2 inches by 12 inches. What is its surface area? Show your reasoning. If you get stuck, consider drawing a sketch of the box or its net and labeling the edges with their measurements.

Exercise \(\PageIndex{6}\)

Twelve cubes are stacked to make this figure.

- What is its surface area?
- How would the surface area change if the top two cubes are removed?

(From Unit 1.5.1)

Exercise \(\PageIndex{7}\)

Here are two polyhedra and their nets. Label all edges in the net with the correct lengths.

Exercise \(\PageIndex{8}\)

- What three-dimensional figure can be assembled from the net?

- What is the surface area of the figure? (One grid square is 1 square unit.)

(From Unit 1.5.3)