11.4 Exploring Solids Geometry How many geometric solid can you name?

Geometry 11.4 Exploring Solids Goals Know terminology about solids. Identify solids by type. Use Euler’s Theorem to solve problems. May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Polyhedron A solid that is bounded by polygons. The polygons are faces. An edge is the intersection of two faces. A vertex is the intersection of three or more faces. Face Face Face May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Polyhedron Many Sides May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Polyhedron Views Solid Wire Frame All three views will be used in these presentations, the text and other materials. Hidden Line May 12, 2019 Geometry 11.4 Exploring Solids

Which of these are Polyhedrons? NO YES YES NO YES May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Concave Polyhedra A diagonal, or part of a diagonal, is outside the figure. May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Regular Polyhedra All of the faces are congruent, regular polygons. May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Cross Section The intersection of a solid and a plane. Cross section is a circle. May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Cross Section What is the intersection now? Cross section is a rectangle. May 12, 2019 Geometry 11.4 Exploring Solids

What would the cross section be? A Square May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Leonard Euler 1707 – 1783 One of the greatest mathematician of all time. Worked in, and made enormous contributions to, every branch of mathematics. May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Euler’s Formula Count F, the number of faces. 4 3 1 5 6 F = 6 2 May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Euler’s Formula 7 6 5 8 2 3 Count V, the number of vertices. 1 4 V = 8 May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Euler’s Formula 7 6 10 5 8 11 3 12 9 2 1 4 Count E, the number of edges. E = 12 May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Euler’s Formula Faces =6 Vertices = 8 Edges = 12 V + F = E + 2 May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Euler’s Formula Faces = 6 Vertices = 8 Edges = 12 6 + 8 = 12 + 2 May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Euler’s Formula Faces = 6 Vertices = 8 Edges = 12 6 + 8 = 12 + 2 14 = 14 May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Euler’s Formula V + F = E + 2 May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Try another figure… Faces = Vertices = Edges = F + V = E + 2 5 + 5 = 8 + 2 10 = 10 May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Euler’s Formula V + F = E + 2 May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Solve: A polyhedron has 8 faces and 12 vertices. How many edges does it have? 18 V + F = E + 2 12 + 8 = E + 2 20 = E + 2 E = 18 May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Solve: A polyhedron has 24 vertices and 36 edges. How many faces does it have? 14 V + F = E + 2 24 + F = 36 + 2 24 + F = 38 F = 14 May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Solve: A polyhedron has 32 faces and 60 edges. How many vertices does it have? 30 V + F = E + 2 V + 32 = 60 + 2 V + 32 = 62 V= 30 May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids The Platonic Solids There are only five of them. They are regular, convex polyhedra. First described ca. 350 BC by Plato in Timaeus. Have been found in many ancient cultures. May 12, 2019 Geometry 11.4 Exploring Solids

The Five Platonic Solids May 12, 2019 Geometry 11.4 Exploring Solids

Tetrahedron Has four triangular sides. May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Hexahedron (cube) Has six square sides. May 12, 2019 Geometry 11.4 Exploring Solids

Octahedron Has eight triangular sides. May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Dodecahedron Has 12 pentagonal faces. May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Icosahedron Has 20 triangular faces. May 12, 2019 Geometry 11.4 Exploring Solids

Public Toilets in South Korea This is not a Platonic Solid. It is a compound May 12, 2019 Geometry 11.4 Exploring Solids

Geometry 11.4 Exploring Solids Summary A polyhedron is a solid object. The sides are faces. Regular polyhedra have congruent faces. There are 5 regular polyhedra (the Platonic Solids). Euler’s Formula: F + V = E + 2 May 12, 2019 Geometry 11.4 Exploring Solids