Surface Area and Volume Chapter 12

Exploring Solids 12.1 California State Standards 8, 9: Solve problems involving the surface area and lateral area of geometric solids and COMMIT TO MEMORY THE NECESSARY FORMULAS.

definition 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 vertex Plural: polyhedra

examples Polyhedra Prism Pyramid Non-Polyhedra Cylinder Cone Sphere

definition Regular Polyhedron All faces are congruent regular polygons

definition Convex Polyhedron A polyhedron where any line segment between any two points lies entirely within the polyhedron convex non-convex

definition Platonic Solids 5 regular, convex polyhedra Tetrahedron4 equilateral triangles Hexahedron6 squares Octahedron8 equilateral triangles Dodecahedron12 regular pentagons Icosahedron20 equilateral triangles

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

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

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

theorem 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

example A solid has 10 faces. 4 triangles 1 square 4 hexagons1 octagon. How many vertices does the solid have?

example 4 triangles have 4(3) = 12 sides 1 square has 4 sides 4 hexagon has 4(6) = 24 sides 1 octagon has 8 sides 48 individual sides Each side is shared by two polygons edges in the polyhedron

example F + V – E = V – 24 = V = 2 V = 16

example A soccer ball is made of 12 pentagons and 20 hexagons. How many vertices does the soccer ball have? 12 pentagons have 12(5) = 60 sides 20 hexagons have 20(6) = 120 sides 180 individual sides Each side is shared by two polygons edges in the polyhedron

example F + V – E = V – 90 = V = 2 V = 60