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Exploring 3-D Shapes: Surface Area and Volume of Solids

This chapter discusses various topics related to 3-D shapes, including surface area and volume of solids. It covers concepts such as polyhedra, regular polyhedra, convex and concave polyhedra, cross sections, and Euler's theorem.

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Exploring 3-D Shapes: Surface Area and Volume of Solids

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  1. Chapter 12 Surface Area and Volume

  2. Topics We Will Discuss • 3-D Shapes (Solids) • Surface Area of solids • Volume of Solids

  3. Some Vocab We Should Know • Equilateral Triangle • Polygon • Convex • Nonconvex • Ratio • Scale Factor

  4. Skill Review

  5. Geometry 12.1 Exploring Solids

  6. POLYHEDRA A polyhedron is a solid that is bounded by polygons, called faces, that enclose a single region of space. An edge of a polyhedron is a line segment formed by the intersection of two faces. A vertex of a polyhedron is a point where three or more edges meet.

  7. Example of a Polyhedron

  8. Let’s Explore some 3-D Shapes For each shape that you receive. State whether it is a polyhedron, if it is state how many faces, edges, and vertices it has.

  9. Types of Solids

  10. REGULAR POLYHEDRA A polyhedron is regular if all of its faces are congruent regular polygons.

  11. CONVEX POLYHEDRA A polyhedron is convex if any two points on its surface can be connected by a segment that lies entirely inside or on the polyhedron. If this segment goes outside the polyhedron, then the polyhedron is nonconvex, or concave.

  12. CONCAVE POLYHEDRA

  13. CROSS SECTION Imagine a plane slicing through a solid. The intersection of the plane and the solid is called a cross section.

  14. The 5 Regular PolyhedraPlatonic Solids • Regular Tetrahedron • Cube • Regular Octahedron • Regular Dodecahedron • Regular Icosahedron

  15. Euler’s Theorem The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula F + V = E + 2

  16. Example A solid has 10 faces: 4 triangles, 1 square, 4 hexagons, and 1 octagon. How many vertices does the solid have?

  17. Example A solid has 11 faces: 5 quadrilaterals and 6 pentagons. How many vertices does the solid have?

  18. Example Some quartz crystals are pointed on both ends, and have 14 vertices and 30 edges. If you plan to put a label on one of the faces of a crystal, how many faces do you have to choose from?

  19. Example A paper model of a geodesic dome is composed of 180 triangular faces. How many vertices does it have?

  20. Example Like a soccer ball, a snub dodecahedron has 12 pentagonal faces. The rest of its 92 faces are triangle. How many vertices does the solid have?

  21. Quick Questions • What makes a polyhedron a regular polyhedron? • How can you find the number of vertices of a polyhedron if you know the number of faces and edges?

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