Surface Area and Volume Chapter 12 Surface Area and Volume

Chapter 12 Objectives Define polyhedron Utilize Euler’s Theorem Identify cross sections Identify a prism and cylinder Identify a pyramid and cone Calculate the surface area of special polyhedra Calculate the volume of special polyhedra Identify a sphere Calculate the surface area and volume of a sphere Compare quantities between similar polyhedra

Lesson 12.1 Exploring Solids

Lesson 12.1 Objectives Define polyhedron Use properties of polyhedra Utilize Euler’s Theorem

Polyhedron A polyhedron is a solid made of polygons. Remember, polygons are 2-D shapes with line segments for sides. The polygons form faces, or sides of the solid. An edge of a polyhedron is the line segment that is formed by the intersection of 2 faces. Typically the sides of the polygon faces. A vertex of a polyhedron is a point in which 3 or more edges meet Typically the corners of the polygon faces.

Example of a Polyhedron Notice that a polyhedron is nothing more than a 3-dimensional shape. It has some form of a height, a width, and a depth. Faces Edges Vertices The plural of polyhedron is polyhedra.

Example 1 Determine if the following figures are polyhedra. Explain your reasoning. No Yes No There are no faces. There are no polygons. All faces are polygons. One of the faces is not a polygon.

Regular vs Convex A polyhedron is regular if ALL of its faces are regular polygons. Some polyhedron are regular, some are not! A polyhedron is convex if no line segment connecting two interior points leaves the polyhedron and re- enters. No dents! If there are dents, then the polyhedron is concave.

Cross Sections When you take a plane and cut through a solid, the resulting shape of the surface is called the cross section. When asked to identify a cross section, you need to identify the polygon formed. The plane acts like a knife blade and cuts through the solid.

Theorem 12.1: Euler’s Theorem The number of faces (F), the number of vertices (V), and the number of edges (E) in a polyhedron are related by F + V = E + 2 This is commonly used to find one of the variables above. Typically this is because it is hard to see all the faces, vertices, and edges. 6 + 8 = E + 2 F = 6 14 = E + 2 V = 8 12 = E E = ???

Example 2 Find the number of vertices, faces, and edges each polyhedron has. F + V = E + 2 F = 5 F = 6 F = 8 F + 8 = 12 + 2 V = 6 V = 6 V = 12 F + 8 = 14 5 + 6 = E + 2 11 = E + 2 6 + 6 = E + 2 8 + 12 = E + 2 F = 6 E = 9 12 = E + 2 20 = E + 2 E = 10 E = 18 You do not have to use the formula to find edges every time. You can use it to find any of the missing quantities as long as you know the other 2, if you use the formula at all!

Regular Polyhedra There are only five regular polyhedra, called Platonic solids. Named after Greek mathematician and philosopher Plato They use regular triangles, squares, and regular pentagons to form solids. Those are the only 3 shapes that can be used. That is because when their vertices are butted together, their interior angles can add to 360o.

Platonic Solids Name Face Shape # Faces Vertices Edges Tetrahedron Triangle 4 6 Cube Square 8 12 Octahedron Dodecahedron Pentagon 20 30 Icosahedron

Homework 12.1 In Class 1-9 p723-726 HW 10-55, 60-70 ev Due Tomorrow

Surface Area of Prisms and Cylinders Lesson 12.2 Surface Area of Prisms and Cylinders

Lesson 12.2 Objectives Identify a prism Calculate the surface area of a prism Identify a cylinder Calculate the surface area of a cylinder Construct a two-dimensional net for three- dimensional solids

Prisms A prism is a polyhedron with two congruent faces that are parallel to each other. The congruent faces are called bases. The bases must be parallel to each other. The other faces are called lateral faces. These are always rectangles or parallelograms or squares. When naming a prism, they are always named by the shape of their bases. Triangular Prism

Parts of a Prism The edges that connect opposing bases are called lateral edges. The height of a prism is the perpendicular distance between the bases. In a right prism, the length of the lateral edge is the height. A right prism is one that stands up straight with the lateral edges perpendicular to the bases. In an oblique prism, the height must be drawn in so that it is perpendicular to both bases. An oblique prism is one that is slanted to one side or the other. The length of the slanted lateral edge is called the slant height.

Examples of Prisms base lateral face slant height lateral edge

Example 3 Answer the following questions: What kind of figure is the base? Hexagon What kind of figure is each lateral face? Rectangle How many lateral faces does the figure have? 6 Same number as the sides of the base! Give the mathematical name of the geometrical solid. Hexagonal Prism

Area The surface area of a prism is the sum of the areas of all the faces and bases. The lateral area of a prism is the sum of the areas of the lateral faces.

Theorem 12.2: Surface Area of a Prism The surface area (S) of a right prism can be found using the formula S = 2B + Ph B is area of the base A = ½ aP (any polygon) A = lw (rectangle) A = bh (parallelogram) A = ½ bh (triangle) A = s2 (square) P is perimeter of the base h is height of the prism

Example 4 Find the surface area of the following: ? S = 2B + Ph a2 + b2 = c2 52 + 122 = c2 25 + 144 = c2 169 = c2 c = 13 Example 4 Find the surface area of the following: ? S = 2B + Ph S = 2B + Ph S = 2(5•6) + Ph S = 2(1/2•12•5) + Ph S = 2(5•6) + (5+6+5+6)h S = 2(5•6) + (5+12+ )h 13 S = 2(5•6) + (5+6+5+6)(7) S = 2(5•6) + (30)h S = 2(30) + (22)(7) S = 2(30) + (30)(10) S = 60 + 154 S = 60 + 300 S = 214 m2 S = 360 m2 area

Cylinder A cylinder is a solid with congruent and parallel circles for bases. height

Theorem 12.3: Surface Area of a Cylinder The surface area (S) of a right cylinder is S = 2B + Ch B is area of the base A =  r2 C is circumference of the base C = 2 r or C =  d h is height of the cylinder

Example 5 S = 2B + Ch S = 2r2 + Ch S = 2r2 + 2rh Find the surface area of the following cylinder. Round your answer to the nearest hundreth. S = 2B + Ch S = 2r2 + Ch S = 2r2 + 2rh S = 2(9)2 + 2(9)(7) S = 2(81) + 2(63) S = 162 + 126 S = 288 S = 508.94 + 395.84 = 904.78 ft2

Nets A net is a two-dimensional drawing of a three-dimensional solid. If you were to unfold a solid, the net would show what it looks like. Every solid has a net. However, there are only certain ways to draw a net for each solid.