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 face edge vertex

Warm-Up Separate your Geosolid polyhedra into 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 prism iff 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 A sphere doesn’t have a true net; it can only be approximated. Match one of the red, rubbery nets with its corresponding 3-D solid. Which of the shapes has no net?

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 C D 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