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**Assignment**• P. 806-9: 2-20 even, 21, 24, 25, 28, 30 • P. 814-7: 2, 3-21 odd, 22-25, 30 • Challenge Problems: 3-5, 8, 9**In Glorious 3-D!**Most of the figures you have worked with so far have been confined to a plane—two-dimensional. Solid figures in the “real world” have 3 dimensions: length, width, and height.**Polyhedron**A solid formed by polygons that enclose a single region of space is called a polyhedron. Separate your Geosolids into 2 groups: Polyhedra and others.**Parts of Polyhedrons**• Polygonal region = face • Intersection of 2 faces = edge • Intersection of 3+ edges = vertex edge face vertex**Warm-Up**Separate your Geosolidpolyhedrainto two groups where each of the groups have similar characteristics. What are the names of these groups? Prisms Pyramids Polyhedra:**12.2 & 12.3: Surface Area of Prisms, Cylinders, Pyramids,**and Cones Objectives: • To find and use formulas for the lateral and total surface area of prisms, cylinders, pyramids, and cones**Prism**A polyhedron is a prismiff it has two congruent parallel bases and its lateral faces are parallelograms.**Classification of Prisms**Prisms are classified by their bases.**Right & Oblique Prisms**Prisms can be right or oblique. What differentiates the two?**Right & Oblique Prisms**In a right prism, the lateral edges are perpendicular to the base.**Pyramid**A polyhedron is a pyramid iff it has one base and its lateral faces are triangles with a common vertex.**Classification of Pyramids**Pyramids are also classified by their bases.**Pyramid**A regular pyramid is one whose base is a regular polygon.**Pyramid**A regular pyramid is one whose base is a regular polygon. • The slant height is the height of one of the congruent lateral faces.**Solids of Revolution**The three-dimensional figure formed by spinning a two dimensional figure around an axis is called a solid of revolution.**Cylinder**A cylinder is a 3-D figure with two congruent and parallel circular bases. • Radius = radius of base**Cone**A cone is a 3-D figure with one circular base and a vertex not on the same plane as the base. • Altitude = perpendicular segment connecting vertex to the plane containing the base (length = height)**Cone**A cone is a 3-D figure with one circular base and a vertex not on the same plane as the base. • Slant height= segment connecting vertex to the circular edge of the base**Right vs. Oblique**What is the difference between a right and an oblique cone?**Right vs. Oblique**In a right cone, the segment connecting the vertex to the center of the base is perpendicular to the base.**Nets**Imagine cutting a 3-D solid along its edges and laying flat all of its surfaces. This 2-D figure is a net for that 3-D solid. An unfolded pizza box is a net!**Nets**Imagine cutting a 3-D solid along its edges and laying flat all of its surfaces. This 2-D figure is a net for that 3-D solid.**Activity: Red, Rubbery Nets**Match one of the red, rubbery nets with its corresponding 3-D solid. Which of the shapes has no net?**Activity: Red, Rubbery Nets**Match one of the red, rubbery nets with its corresponding 3-D solid. Which of the shapes has no net? Asphere doesn’t have a true net; it can only be approximated.**Exercise 1**There are generally two types of measurements associated with 3-D solids: surface area and volume. Which of these can be easily found using a shape’s net?**Surface Area**The surface area of a 3-D figure is the sum of the areas of all the faces or surfaces that enclose the solid. • Asking how much surface area a figure has is like asking how much wrapping paper it takes to cover it.**Lateral Surface Area**The lateral surface area of a 3-D figure is the sum of the areas of all the lateral faces of the solid. • Think of the lateral surface area as the size of a label that you could put on the figure.**Exercise 2**What solid corresponds to the net below? How could you find the lateral and total surface area?**Exercise 3**Draw a net for the rectangular prism below. A B D C To find the lateral surface area, you could: • Add up the areas of the lateral rectangles**Exercise 3**Draw a net for the rectangular prism below. Height of Prism Perimeter of the Base To find the lateral surface area, you could: • Find the area of the lateral surface as one, big rectangle**Exercise 3**Draw a net for the rectangular prism below. Height of Prism Perimeter of the Base To find the total surface area, you could: • Find the lateral surface area then add the two bases**Surface Area of a Prism**Lateral Surface Area of a Prism: • P = perimeter of the base • h = height of the prism Total Surface Area of a Prism: • B = area of the base**Exercise 4**Find the lateral and total surface area.**Exercise 5**Draw a net for the cylinder. Notice that the lateral surface of a cylinder is also a rectangle. Its height is the height of the cylinder, and the base is the circumference of the base.**Exercise 6**Write formulas for the lateral and total surface area of a cylinder.**Surface Area of a Cylinder**Lateral Surface Area of a Cylinder: • C = circumference of base • r = radius of base • h = height of the cylinder Total Surface Area of a Cylinder:**Exercise 7**The net can be folded to form a cylinder. What is the approximate lateral and total surface area of the cylinder?**Height vs. Slant Height**By convention, h represents height and l represents slant height.**Height vs. Slant Height**By convention, h represents height and l represents slant height.**Exercise 8**Draw a net for the square pyramid below. To find the lateral surface area: • Find the area of one triangle, then multiply by 4**Exercise 8**Draw a net for the square pyramid below. To find the lateral surface area: • Find the area of one triangle, then multiply by 4**Exercise 8**Draw a net for the square pyramid below. To find the total surface area: • Just add the area of the base to the lateral area**Surface Area of a Pyramid**Lateral Surface Area of a Pyramid: • P = perimeter of the base • l = slant height of the pyramid Total Surface Area of a Prism: • B = area of the base**Exercise 9**Find the lateral and total surface area.**Exercise 10**You may have realized that the formula for the lateral area for a prism and a cylinder are basically the same. The same is true for the formulas for a pyramid and a cone. Derive a formula for the lateral area of a cone. Lateral area of a Pyramid: Lateral area of a Cone:**Surface Area of a Cone**Lateral Surface Area of a Cone: • r = radius of the base • l = slant height of the cone Total Surface Area of a Cone:**Exercise 11**A traffic cone can be approximated by a right cone with radius 5.7 inches and height 18 inches. To the nearest tenth of a square inch, find the approximate lateral area of the traffic cone.**Tons of Formulas?**Really there’s just two formulas, one for prisms/cylinders and one for pyramids/cones.**Assignment**• P. 806-9: 2-20 even, 21, 24, 25, 28, 30 • P. 814-7: 2, 3-21 odd, 22-25, 30 • Challenge Problems: 3-5, 8, 9