Lesson 12
More Nets, More Surface Area
Let’s draw nets and find the surface area of polyhedra.
12.1: 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.](https://cms-im.s3.amazonaws.com/S4pMvqUH6Z1z3ae4inkFNbrH?response-content-disposition=inline%3B%20filename%3D%226-6.1.E4.Image.01a.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.E4.Image.01a.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T042952Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=3349cfa38818861c48560da88ca4fd7b0aecc2570496b6e20ac84f78dabe0b2b)
What do you notice? What do you wonder?
12.2: Building Prisms and Pyramids
Your teacher will give you a drawing of a polyhedron. You will draw its net and calculate its surface area.
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What polyhedron do you have?
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Study your polyhedron. Then, draw its net on graph paper. Use the side length of a grid square as the unit.
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Label each polygon on the net with a name or number.
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Find the surface area of your polyhedron. Show your thinking in an organized manner so that it can be followed by others.
12.3: 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.](https://cms-im.s3.amazonaws.com/XcCkwzWdEAybnbL7G4LStmD4?response-content-disposition=inline%3B%20filename%3D%226-6.1.E4.Image.04a.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.E4.Image.04a.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T042952Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=6103586d4d55e7a8b6a9a9e7cf57ad14ce8a2c9ae919d28ce00569a9dfb0d924)
- 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.
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.](https://cms-im.s3.amazonaws.com/Jbu3fXMdmNTMCK7wYsxJy6jo?response-content-disposition=inline%3B%20filename%3D%226-6.1.E4_Image_8.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.E4_Image_8.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T042952Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=727c9fc3503989b8ea5c69fc5e2a0428f15af9765e6ce9209c537cdd58ff3fd0)
![A net with a square of side length 6 surrounded by triangles of height 5.](https://cms-im.s3.amazonaws.com/st37zFNLv9sqUMxSZiaqAorR?response-content-disposition=inline%3B%20filename%3D%226-6.1.E4_Image_9.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.E4_Image_9.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T042952Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f23607109e77a7f0d400178d848710c71dfc18f6ff64cdaa8e938ac38cbab82a)
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.
- 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.
- 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.
- 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.