Lesson 14
Nets and Surface Area
Let’s use nets to find the surface area of polyhedra.
14.1: Matching Nets
Each of the nets can be assembled into a polyhedron. Match each net with its corresponding polyhedron, and name the polyhedron. Be prepared to explain how you know the net and polyhedron go together.
![Five nets of polyhedra labeled 1--5.](https://cms-im.s3.amazonaws.com/pVT1LjL5rUoZJTnLLcwRQKGF?response-content-disposition=inline%3B%20filename%3D%226-6.1.E3_Image_2.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.E3_Image_2.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=20240727T004318Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a799f5f91642101ee59b9f2e30ea590ad1848d7ef3a40bcf3032958822e7c4d5)
![Five polyhedra labeled A--E.](https://cms-im.s3.amazonaws.com/sAzq5Ab7DSnQQm3YqyF7QYmu?response-content-disposition=inline%3B%20filename%3D%226-6.1.E3_Image_2.1.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.E3_Image_2.1.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=20240727T004318Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=90bcdd7ba20e9fe168d9303b93fa6e730ee30c55a93a87ec3c3f89b59aee74d2)
14.2: Using Nets to Find Surface Area
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Name the polyhedron that each net would form when assembled.
- Your teacher will give you the nets of three polyhedra. Cut out the nets and assemble the three-dimensional shapes.
- Find the surface area of each polyhedron. Explain your reasoning clearly.
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For each net, decide if it can be assembled into a rectangular prism.
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For each net, decide if it can be folded into a triangular prism.
Summary
A net of a pyramid has one polygon that is the base. The rest of the polygons are triangles. A pentagonal pyramid and its net are shown here.
![The net for this pentagonal pyramid is a pentagon surrounded by triangles on each side.](https://cms-im.s3.amazonaws.com/s6g27fuRAV7VKfDvEM2hJvgc?response-content-disposition=inline%3B%20filename%3D%226-6.1.E3_Image_9.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.E3_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=20240727T004318Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=88292558b60ed0e5ff70d1b44b43d1359b4ca3d84dd69e7b270fcd11f7739481)
A net of a prism has two copies of the polygon that is the base. The rest of the polygons are rectangles. A pentagonal prism and its net are shown here.
![The net for this pentagonal prism is a pentagon surrounded by rectangles on each side with an additional pentagon attached to the opposite side of one of the rectangles.](https://cms-im.s3.amazonaws.com/B4QS87DVvoU4mYeUrax9i6HC?response-content-disposition=inline%3B%20filename%3D%226-6.1.E3_Image_8.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.E3_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=20240727T004318Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=9718360a6a7e39917f0d9c445452961b1492f44b9cd9090db98d8a17e5c3ae83)
In a rectangular prism, there are three pairs of parallel and identical rectangles. Any pair of these identical rectangles can be the bases.
![Three images of a rectangular prism. Each image has one set of opposing sides of the polyhedron shaded and labeled “base."](https://cms-im.s3.amazonaws.com/9zBMNwWRkU4tbW7N9drsBwWV?response-content-disposition=inline%3B%20filename%3D%226-6.1.E3.Image.00.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.E3.Image.00.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=20240727T004318Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=404d82caec915ec5f6612d48625d12101d055d422080ffd808f415f76126d70e)
Because a net shows all the faces of a polyhedron, we can use it to find its surface area. For instance, the net of a rectangular prism shows three pairs of rectangles: 4 units by 2 units, 3 units by 2 units, and 4 units by 3 units.
![A polyhedron made up of six rectangles. Two rectangles are 8 square units in area, 2 are 6 square units, and 2 are 12 square units.](https://cms-im.s3.amazonaws.com/QCuqjYrgRH3QmiyvRThQGunf?response-content-disposition=inline%3B%20filename%3D%226-6.1.E3.Image.07aa.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.E3.Image.07aa.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=20240727T004318Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=3cc3119bebfcbfa2867bfa1b8982d28e43635f3d91b09907e263dd29f84b1b22)
The surface area of the rectangular prism is 52 square units because \(8+8+6+6+12+12=52\).
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.