11.4 Three-Dimensional Figures Geometry 11.4 Three-Dimensional Figures mbhaub@mpsaz.org

Geometry 11.4 Three-Dimensional Figures Warmup May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures Essential Question What is the relationship between the numbers of vertices V, edges E, and faces F of a polyhedron May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures Goals Classify solids Describe cross sections Sketch and describe solids of revolution May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures 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 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures Polyhedron Many Sides May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures 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 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures Types of Solids May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures Naming Polyhedra To name a prism or a pyramid, use the shape of the base. The two bases of a prism are congruent polygons in parallel planes. The base of a pyramid is a polygon. May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Name each solid. Which of these are Polyhedrons? Pentagonal Prism YES Cylinder NO Triangular Prism YES Square Pyramid YES Sphere NO May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures Concave Polyhedra A diagonal, or part of a diagonal, is outside the figure. May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures Regular Polyhedra All of the faces are congruent, regular polygons. May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures Cross Section The intersection of a solid and a plane. The cross section is parallel to the bases. Cross section is a circle. May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures Cross Section What is the intersection now? Cross section is a rectangle. The cross section is perpendicular to the bases. May 12, 2019 Geometry 11.4 Three-Dimensional Figures

What would the cross section be? The cross section is parallel to the base. A Square May 12, 2019 Geometry 11.4 Three-Dimensional Figures

What would the cross section be? The cross section is perpendicular to both bases. A Pentagon May 12, 2019 Geometry 11.4 Three-Dimensional Figures

What would the cross section be? The cross section is slanted and intersects both bases. A Trapezoid May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures 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 Three-Dimensional Figures

The Five Platonic Solids May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Tetrahedron Has four triangular sides. Associated with fire. May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures Hexahedron (cube) Has six square sides. Associated with earth. May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Octahedron Has eight triangular sides. Associated with air. May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Dodecahedron Has 12 pentagonal faces. Associated with the heavens. May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures Icosahedron Has 20 triangular faces. Associated with water. May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures Johannes Kepler In 1596 Kepler published a tract called The Cosmic Mystery in which he envisioned the universe as consisting of nested Platonic Solids whose inscribed spheres determine the orbits of the planets, all enclosed in a sphere representing the outer heavens. May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures Dungeons and Dragons May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Public Toilets in South Korea This is not a Platonic Solid. It is a compound polyhedron. Can you find out its correct name? May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures A solid of revolution is a three-dimensional figure that is formed by rotating a two-dimensional shape around an axis. If you rotate a rectangle around one of its sides, the path it makes through space is a cylinder. May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Solids of Revolution Example: Sketching a Solid of Revolution Draw the solid of revolution formed by the shape rotated around the axis given. Describe the resulting shape. A cylinder with the center removed. May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures Check it Out! Example 1 Draw the solid of revolution formed by the shape rotated around the axis given. Describe the resulting shape. A donut shape, also known as a torus. May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Find the axis of symmetry. Example 2: Sketching a Solid of Revolution Draw a two-dimensional shape and axis of rotation that could form each figure. Find the axis of symmetry. The two-dimensional shape should match the outline of one side of the shape. May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures Check it Out! Example 2 Draw a two-dimensional shape and axis of rotation that could form the sports drink bottle. Find the axis of symmetry. The two-dimensional shape should match the outline of one side of the shape. May 12, 2019 Geometry 11.4 Three-Dimensional Figures

Geometry 11.4 Three-Dimensional Figures 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: May 12, 2019 Geometry 11.4 Three-Dimensional Figures