11.1: What are Polyhedra?
Here are pictures that represent polyhedra:
Here are pictures that do not represent polyhedra:
Your teacher will give you some figures or objects. Sort them into polyhedra and non-polyhedra.
- What features helped you distinguish the polyhedra from the other figures?
11.2: Prisms and Pyramids
- Here are some polyhedra called prisms.
Here are some polyhedra called pyramids.
- Look at the prisms. What are their characteristics or features?
- Look at the pyramids. What are their characteristics or features?
- Which of these nets can be folded into Pyramid P? Select all that apply.
Your teacher will give your group a set of polygons and assign a polyhedron.
- Decide which polygons are needed to compose your assigned polyhedron. List the polygons and how many of each are needed.
Arrange the cut-outs into a net that, if taped and folded, can be assembled into the polyhedron. Sketch the net. If possible, find more than one way to arrange the polygons (show a different net for the same polyhedron).
What is the smallest number of faces a polyhedron can possibly have? Explain how you know.
11.3: Using Nets to Find Surface Area
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.
For each net, decide if it can be assembled into a rectangular prism.
For each net, decide if it can be folded into a triangular prism.
A prism is a type of polyhedron with two identical faces that are parallel to each other and that are called bases. The bases are connected by a set of rectangles (or sometimes parallelograms). A prism is named for the shape of its bases. For example, if the base is a pentagon, then it is called a “pentagonal prism.”
A net is a two-dimensional representation of a polyhedron. It is composed of polygons that form the faces of a polyhedron. 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.
A pyramid is a type of polyhedron that has one special face called the base. All of the other faces are triangles that all meet at a single vertex. A pyramid is named for the shape of its base. For example, if the base is a pentagon, then it is called a “pentagonal pyramid.”
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.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. The surface area of the rectangular prism is 52 square units because \(8+8+6+6+12+12=52\).
- 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.
Each flat side of a polyhedron is called a face. For example, a cube has 6 faces, and they are all squares.
A net is a two-dimensional figure that can be folded to make a polyhedron.
Here is a net for a cube.
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.
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 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.