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Warm up

Explore Euclid's Postulates, solid geometry, and the properties of polyhedrons, faces, vertices, and more. Learn about Platonic and Kepler-Poinsot solids, nets, and cross-sections in this comprehensive guide.

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Warm up

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  1. Warm up State which of Euclid’s Postulates is being used. State whether each statement is True or False. If false, state why. • Any line segment may be extended indefinitely to form a line. • Given a line, a circle can be drawn having the segment as a radius and one endpoint as a center • If l is any line and P is any point not on l, then there exists exactly one line through P that is parallel to l. • A line segment can be drawn joining any three points • All congruent angles are right angles

  2. Answers • Postulate 2, True • Postulate 3, False it is a line segment • Postulate 5, True • Postulate 1, False only two points are needed • Postulate 4, False all right angles are congruent

  3. Shape Geometry4.1.2: Visualize solids and surfaces in 3-dimensional space when given 2-dimensional representations and create 2-dimensional representations for the surfaces of 3-dimensional objects

  4. face base Some basic vocabulary you should know! A Polyhedron is a solid formed by polygons that enclose a single region of space. A face is the flat polygonal surfaces of the polyhedron In a solid prism there will be a maximum of 2 identical faces called bases.

  5. vertex edge base Lateral face Words to know continued… A lateral face is the polygonal faces that are not bases. An edge is a segment where two faces intersect. A vertex is the point where three or more edges meet.

  6. A couple more… You name prisms and pyramids by the name of their base. A prism is special type of polyhedron, with 2 faces called bases, that are congruent, parallel polygons. Let’s draw some prisms

  7. Hang in there… A pyramid is special type of polyhedron they have only one base, the other faces are lateral faces that meet to form the lateral edges. The common vertex of the lateral faces is the vertex of the pyramid. Pyramids are classified by their bases

  8. But… what about cylinders, cones and spheres? Cylinders, cones and spheres are also geometric solids. They have curved surface. A cone has a circle base, the vertex is perpendicular to the base A cylinder has two bases that are circles. A sphere is the set of all points at a given distance from a given point (center).

  9. Almost done… When a solid is cut by a plane the resulting two-dimensional figure is called a cross-section. A net is a two-dimensional pattern that you can cut and fold to form a three-dimensional figure.

  10. Investigate 3-D shapes In groups of three get the following supplies: • 1 box of floss • 1 mound or jar of clay • 1 lab write-up • 1 set of nets

  11. Part 1: Investigating faces Part 2: investigating Nets • Each group will build a different solid. • Use the lab sheet for directions. • Include all drawings on the lab write-up • Be prepared to share with the class.

  12. Platonic Solids • A Platonic solid is a polyhedron all of whose faces are congruent regular convex polygons*, and where the same number of faces meet at every vertex.

  13. Why are there only five Platonic Solids • The Greeks recognized that there are only five platonic solids. But why is this so? • The key observation is that the interior angles of the polygons meeting at a vertex of a polyhedron add to less than 360 degrees.

  14. Platonic Solids • Tetrahedron:Three triangles at a vertex: 3*60 = 180 degrees • Octahedron: Four triangles at a vertex: 4*60 = 240 degrees • Icosahedron:Five triangles at a vertex: 5*60 = 300 degrees • Cube: Three squares at a vertex: 3*90 = 270 degrees • Dodecahedron: Three pentagons at a vertex: 3*108 = 324 degrees • Note:  Six triangles: 6*60 = 360 degrees  Four squares: 4*90 = 360 degrees  Four pentagons: 4*108 = 432 degrees  Three hexagons: 3*120 = 360 degrees So there are only five Platonic Solids!

  15. Dodecahedron Octahedron Icosahedron Cube Tetrahedron

  16. Regular Solids • There are nine regular solids: the five Platonian and the four polyhedra described by Kepler-Poinsot. • Each face, apex and angle on each respective solid is the same.

  17. Kepler-Poinsot Solids The four regular non-

  18. Visualizing solids • http://illuminations.nctm.org/ActivityDetail.aspx?id=70 Summary: What is a net? How can we use nets to better understand figures? What do cross-sections tell us about the figure?

  19. Homework 1.8 page pg 78 # 19-22 AND page 83 # 1-6,13-17, 20-21

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