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**Surface Area and Volume**

Chapter 12

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**Exploring Solids 12.1 California State Standards Lesson goals**

8, 9: Solve problems involving the surface area and lateral area of geometric solids and COMMIT TO MEMORY THE NECESSARY FORMULAS. Identify Polyhedrons Apply Euler’s Theorem

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**definition Polyhedron edge face face vertex**

Plural: polyhedra Polyhedron A 3-dimensional solid figure formed by polygons. Faces: the polygons that form the polyhedron Edges: a line segment formed by the intersection of two faces. Vertex: the point where 3 or more edges meet. face edge face vertex

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examples Polyhedra Prism Pyramid Non-Polyhedra Cylinder Cone Sphere

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definition Regular Polyhedron All faces are congruent regular polygons

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**definition Platonic Solids 5 regular, convex polyhedra**

Tetrahedron 4 equilateral triangles Hexahedron 6 squares Octahedron 8 equilateral triangles Dodecahedron 12 regular pentagons Icosahedron 20 equilateral triangles

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**tetrahedron 4 faces hexahedron 6 faces octahedron 8 faces icosahedron**

(cube) octahedron 8 faces icosahedron 20 faces dodecahedron 12 faces

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**definition Cross-section The intersection of a plane and a solid.**

Describe the cross section. The cross section is a square.

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**definition Cross-section The intersection of a plane and a solid.**

Describe the cross section. The cross section is a circle.

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**theorem Euler’s Theorem The number of faces (F), vertices (V), and**

edges (E) of a polyhedron are related by the formula

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example A polyhedron has 18 edges and 12 vertices. How many faces does it have?

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**Each side is shared by two polygons**

example Find the number of vertices for a polyhedron with 10 faces made from 4 triangles, 1 square, 4 hexagons, and 1 octagon. Each side is shared by two polygons

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**Each side is shared by two polygons**

example A soccer ball is made of 12 pentagons and 20 hexagons. How many vertices does the soccer ball have? Each side is shared by two polygons

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Definition Net the two-dimensional representation of a three-dimensional figure. back left roof right left side right bottom front What would the polyhedron look like if laid flat?

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**Remember: this is called a NET.**

example What would the cylinder look like laid flat? top “label” bottom Remember: this is called a NET.

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Today’s Assignment p. 723: 6 – 15, 25 – 31, 47 – 49

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