Solid Figures Section 9.1

Goal: Identify and name solid figures.

Key Vocabulary Solid Polyhedron Face Edge Base

Introduction to Three-Dimension Figures Three-dimensional figures are not flat figures. They have length, width, and height. They are also called solid geometric figures.

Introduction to Three-Dimension Figures A flat surface of a solid is a face. An edge is where two faces meet. A vertex is where three or more edges meet. The face that is used to classify a solid is a base. face base vertex edge

Types of Solids The surfaces of a three-dimensional figure determine the type of solid it is. A polyhedron is a three- dimensional figure whose surfaces, or faces, are all polygons. Prisms and pyramids are two types of polyhedrons.

Properties of Polyhedra A polyhedron is a solid bounded by polygons called faces which enclose a single region of space. An edge of a polyhedron is a line segment formed by the intersection of two faces. A vertex of a polyhedron is a point where three or more edges meet. The plural of polyhedron is either polyhedra or polyhedrons.

Identifying Polyhedra Decide whether each of the solids below is a polyhedron. If so, count the number of faces, vertices, and edges of the polyhedron.

a.This is a polyhedron. It has 5 faces, 6 vertices, and 9 edges. b.This is not a polyhedron. Some of its faces are not polygons. c.This is a polyhedron. It has 7 faces, 7 vertices, and 12 edges.

Summary Types of Solids The prism has two parallel congruent faces called bases. The other faces are parallelograms… the pyramid has all but one of its faces (the base) intersecting at one vertex… Both solids are named by the shape of their bases followed by “prism” or “pyramid” as their surname (i.e. triangular prism and square pyramid).

Prisms A prism is a polyhedron that has two parallel congruent bases. The bases can be any polygon. The other faces are parallelograms. A cube is a special prism whose faces are all congruent squares.

Cube edge vertex face A cube, a rectangular prism, has 6 faces (all squares).

Pyramids A pyramid is a polyhedron that has one base. The base can any polygon. The other faces are triangles. A regular tetrahedron is a special pyramid whose faces are all congruent equilateral triangles.

Tetrahedron A pyramid with four faces which are all equilateral triangles. Faces4 Vertices 4 Edges6

Polyhedrons or Polyhedra (prisms and pyramids) are named by the shapes of their bases.

Triangular Prism A prism with triangular parallel bases. A triangular prism has five faces. Its base is a triangle. Two of its faces are triangles; three of its faces are rectangles. It has six vertices and nine edges.

Rectangular Prism A prism with rectangular parallel bases. Also called a Cuboid. A rectangular prism has six faces. Its base is a rectangle. All of its faces are rectangles. It has eight vertices and twelve edges.

Hexagonal Prism A prism with hexagonal parallel bases. A hexagonal prism has eight faces. Its base is a hexagon. It has six faces that are rectangles and two faces that are hexagons. It has twelve vertices and eighteen edges.

Square (or rectangular) Pyramid A shape with a square base and triangular sides that meet at a point. Faces 5 Vertices 5 Edges8

Identify the base or bases of the solid. Then name the solid. Example: Naming Prisms and Pyramids There are two bases, and they are both octagons. The other faces are parallelograms. The figure is an octagonal prism.

Identify the base or bases of the solid. Then name the solid. Example: Naming Prisms and Pyramids There is one base, and it is a pentagon. The other faces are triangles. The figure is a pentagonal pyramid.

Identify the base or bases of the solid. Then name the solid. Try This: Example There are two bases and they are both triangles. The other faces are parallelograms. The figure is a triangular prism.

Identify the base or bases of the solid. Then name the solid. Try This: Example All faces are congruent squares. The figure is a cube.

More Practice

What is the name of a three-dimensional figure with six square faces? a.pyramid b.cube c.solid d.edge

I am a three-dimensional figure with five faces. My base is a rectangle; my other faces are triangles. I have five vertices and eight edges. What am I? a.rectangle prism b.rectangular pyramid c.triangular prism d.triangular pyramid

How many edges does a square pyramid have? a.4 b.5 c.7 d.8

How many vertices does a rectangular prism have? a.12 b.8 c.6 d.4

Which of the following has a base that is a triangle? a.Triangular prism b.Cube c.Square pyramid d.Rectangular pyramid

I am a three-dimensional figure with four faces. All of my faces, including the base are triangles. What am I? a.rectangular prism b.triangular prism c.triangular pyramid d.rectangular pyramid

How many faces of a cube are squares? a.2 b.4 c.6 d.8

Sally is going to build a table for her family room. It is in the shape of a triangular prism. What shapes will she need to build the table? a.5 triangles b.2 triangles and 3 rectangles c.2 triangles and 4 rectangles d.6 rectangles

Not Polyhedrons Other three-dimensional figures include cylinders, cones, and spheres. These figures are different from polyhedrons because they each have a curved surface and their bases are not polygons. The curved surface of a cylinder or a cone is called a lateral surface.

Cylinder A prism with a circular cross-section. The normal definitions of faces, corners and edges are not appropriate for a cylinder. A cylinder has two parallel, congruent circular bases connected by a lateral surface. 2 bases Lateral surface

Cone The point of the cone is directly above the centre of the circular base. A cone has one circular base and a lateral surface. The lateral surface of a cone comes to a point called its vertex. 1 base Lateral surface Vertex

Sphere A sphere has only one surface, which is curved, and has no base. All of the points on the surface are the same distance from the center of the sphere. A plane that intercepts a sphere through its center divides the sphere into two different halves, or hemispheres. Top hemisphere Bottom hemisphere

Example Determine whether the solid is a polyhedron. Then identify the solid. If it is a polyhedron, name the bases, faces, edges, and vertices.

Example The solid has a curved surface, so it is not a polyhedron. The base is a circle and there is one vertex. So, it is a cone. Answer: Base: circle T Vertex: W no faces or edges

Example A.cone B.cylinder C.pyramid D.polyhedron Identify the solid.

Practice Identify the type of each prism or pyramid. pentagonal prism rectangular pyramid cylinder cone 1.2. Identify the figure described. 3. two congruent circular faces connected by a curved surface 4. one circular face and a curved lateral surface that comes to a point called a vertex

Summary

Identify and Name Polyhedra Example 1 Tell whether the solid is a polyhedron. If so, identify the shape of the bases. Then name the solid. SOLUTION a.The solid is formed by polygons so it is a polyhedron. The bases are congruent triangles in parallel planes. This figure is a triangular prism. b.A cylinder has a curved surface, so it is not a polyhedron. a. b.

Find Faces and Edges Example 2 Use the diagram at the right. Name the polyhedron.a. Count the number of faces and edges.b. List any congruent faces and congruent edges.c. SOLUTION The polyhedron is a hexagonal pyramid.a. The polyhedron has 7 faces and 12 edges. b. Using the markings on the diagram, you can conclude the following: c.

Find Faces and Edges Example 2 Congruent faces ∆PQV  ∆QRV  ∆RSV Congruent edges QRPQSTRS TU      UP ∆STV  ∆TUV  ∆UPV PVQVSVRV TV      UV

Sketch a Polyhedron Example 3 Sketch a triangular prism. SOLUTION Draw the triangular bases.1. Connect the corresponding vertices of the bases with vertical lines. 2. Partially erase the hidden lines to create dashed lines. Shade the prism. 3.

Your Turn: Tell whether the solid is a polyhedron. If so, identify the shape of the base(s). Then name the solid ANSWER yes; triangular; triangular prism ANSWER no; cone ANSWER yes; rectangular; rectangular pyramid

Your Turn: Copy the partial drawing of a triangular pyramid. Then complete the drawing of the pyramid. 4. ANSWER

Euler’s Formula Mathematician Leonhard Euler proved that the number of faces (F), vertices (V), and edges (E) of a polyhedron are related. Leonard Euler

Use Euler’s Formula to find the number of edges on a solid with 6 faces and 8 vertices. F + V = E + 2Euler’s Formula = E + 2Substitute the number of faces and vertices. 12 = ESimplify. A solid with 6 faces and 8 vertices has 12 edges. Using Euler’s formula

Regular Prisms and Regular Polyhedra If the bases of a prism are regular polygons, then it is called a regular prism. If all of the faces of a polyhedron are regular congruent polygons and all of the edges are congruent then it is called a regular polyhedron.

Platonic Solids There are five (5) regular polyhedra called Platonic Solids, named after the Greek mathematician and philosopher, Plato. The five Platonic Solids are: 1.Tetrahedron (4 equilateral triangle faces) 2.Hexahedron (6 square faces) 3.Octahedron (8 equilateral triangle faces) 4.Dodecahedron (12 regular pentagon faces) 5.Icosahedron (20 equilateral triangle faces) Tetrahedron Hexahedron Octahedron Dodecahedron Icosahedron

Summary

Review

Homework Pg. 476 – 480: #1 – 9 all, 11 – 23 odd, 25 – 30 odd, 31 – 67 odd