Surface Area and Volume Chapter 12

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

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

examples Polyhedra Prism Pyramid Non-Polyhedra Cylinder Cone Sphere

definition Regular Polyhedron All faces are congruent regular polygons

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

tetrahedron 4 faces hexahedron 6 faces octahedron 8 faces icosahedron (cube) octahedron 8 faces icosahedron 20 faces dodecahedron 12 faces

definition Cross-section The intersection of a plane and a solid. Describe the cross section. The cross section is a square.

definition Cross-section The intersection of a plane and a solid. Describe the cross section. The cross section is a circle.

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

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

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

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

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?

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

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