Unit 8: Polyhedra and solids of revolution
In this lesson you will learn about: Elements of a polyhedra Prisms, parallelelipeda, pyramids and their characteristics. Cylinders, cones, spheres and their characteristcs Vocabulary you need in this lesson: You are going to use some vocabulary related to this lesson and the problems you will work with. The same words appe- ar in the crossword, in the word scramble and in the word search. Some theory
A polyhedron is a solid made of flat surfaces. Each surface is a polygon.
Regular polyhedra:
Tetrahedron Cube Octahedron Dodecahedron
Icosahedron
Prism: an n-side prism is a polyhedron made of n-side polygonal base, a translated copy, and n faces joining corresponding sides. Thus these joining faces are parallelograms.
Pyramid:is a polyhedron formed by connecting a polygonal base and a point called the apex.
Each polyhedron has sides called faces, edges which connect the faces, and vertices or corners which connect the edges. Solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis) that lies on the same plane. The main solids of revolution are: cylinder, cone and sphere.
Cylinder: is a geometrical shape formed by making a rectangle revolve around one of its sides
Cone: is a geometrical shape formed by making a right-angled triangle revolve around one of its cathetus
Sphere: is a geometrical shape formed by making a semicircle revolve around its diameter Now it´s time to practise
Exercises and Problems
1. Write the names of these polyhedra:
2. Draw the following polyhedra and mark the main elements on them: A regular triangular prism. A regular hexagonal prism. A square pyramid.
3. Draw the net (plane development) for the shapes in exercise 2, and identify the ele- ments. 4. Can you recognize the shapes with these nets? 5. Write the name of these solids:
6. Draw the net for a cylinder with a radius of 1.5 cm and a height of 2.5 cm, and mark the measures on it. 7. Draw the net for a cone with a radius of 2 cm and a slant height (generatriz) of 3.5 cm. 8. A rectangle with sides of 7 cm and 5 cm rotates around its shortest side. What kind of solid do you get? 9. Write the name of the solid you get when these shapes rotate around the following axes: A right triangle around one of the legs. An semicircle around its diameter. A square around one of its sides. A right trapezium around its height.
10. A polyhedron is PLATONIC when it obeys the following rules: It is convex. All its faces are equal: squares, triangles, … All its edges are the same length. Complete the table on the next page. How many polyhedra are platonic? What are their names? Look at the last column of the table. What happens? Have you discovered a formula for the polyhedra?
POLYHE- PLA- FACES VER- EDGES (E) F + V - E = DRON TONIC (F) TEXES (V) 6 + 8 – 12 Cube Yes 6 12 8 = 2 Some interesting websites!
If you want to read more about polyhedra and Euler´s Formula go to this page: http://www.mathsisfun.com/geometry/polyhedron.html
To read a little bit about prisma click here: http://www.mathsisfun.com/geometry/prisms.html
To see a complete list of polyhedra go to :
http://www.korthalsaltes.com/
To practice with polyhedra and solids of revolution and check what you have learnt go to thid page: http://clic.xtec.cat/db/act_es.jsp?id=1308
To read a bit more about solids and practice revolving them and observe them in detail click here: http://www.mathsnet.net/geometry/solid/index.html
The formula you have seen in exercise number 10 was discover by Euler, a Swiss mathematician. If you want to read more about this, click here: http://en.wikipedia.org/wiki/Euler_characteristic