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GEOMETRY LESSON 9

SURFACE AREAAND

VOLUME

Jan 30­8:56 PM

AGENDAHomework, review, questions???Objectives:

1. Surface Area and Volume2. Prisms and Cylinders Surface Area3. Nets4. Link Nets to Surface Area5. Volume of Prisms and Cylinders6. Pyramids and Cones

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Discuss the reading from Beckmann: Length, Area, Volume, and Dimension

Use the box you brought to class:

~Describe the one-dimensional parts or aspects of the box.

~Describe the two-dimensional parts or aspects of the box.

~Describe the three-dimensional parts or aspects of the box.

Mar 21­11:07 AM

Representations of Three-Dimensional Figures

To draw three-dimensional figures on two-dimensional paper, we use isometric dot paper and draw isometric views, or corner views of these 3D figures.

Use isometric dot paper to sketch a triangular prism 3 units high with two sides of the base that are 2 units long and 4 units long.

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Answer

Use isometric dot paper and the orthographic drawing to sketch a solid.

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Objective 1. Surface Area and Volume

In your group, define surface area and volume.

Create a list of when you would need a surface area measurement and when you would need to measure volume.

(Note: Beckmann p. 570 defines volume)

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Objective 2: Nets What is a net?Beckmann Activity Manual: 13g-13i

Visualize the shapes that patterns 1,2,3, and 4 on the handout will make. what shapes are they?Compare them and note similarities. Which shapes will be similar, and how will these shapes be similar?

Jan 30­9:36 PM

Beckmann hardcover pp. 563-564--prac ex. (use nets from the back of book)1. Describe each of the following shapes: prisms, cylinders, pyramids, and cones.

2. Try to visualize a (right) prism that has hexagonal bases. How many faces, edges, and vertices does such a prism have?

3. Try to visualize a pyramid that has an octagonal base. How many faces, edges, and vertices does such a pyramid have?

4. What other name can you call a tetrahedron?

5. What happens if you try to make a convex polyhedron whose faces are all equilateral triangles and for which 6 triangles come together at every vertex?6. Is it possible to make a convex polyhedron so that 7 or more equilateral triangles come together at every point? Explain.

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3. Prisms and Cylinders Surface Area

How does oblique apply to both prisms and cylinders?

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Objective 4: Link Nets to Surface Area

Using your boxes, derive surface area formulas for rectangular prisms as you find the surface area of your box.

The following website is a great graphic to use for volume and surface area.

http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/?version=1.5.0_13&browser=Mozilla&vendor=Apple_Computer,_Inc

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Alternative Activity: Practice with surface area.

http://www.aaastudy.comgeo.htm

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Surface Area of Prisms and Cylinders

In a solid figure, faces that are not bases are called lateral faces. The lateral faces intersect each other at the lateral edges, which are parallel and congruent. The lateral faces intersect the base at the base edges. The altitude is a perpendicular segment that joins the planes of the bases. The height is the length of the altitude.

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The lateral area L of a prism is the sum of the areas of the lateral faces. The net below shows how to find the lateral area of a prism.

The lateral area L of a right prism is L = Ph, where h is the height of the prism and P is the perimeter of a base.

Surface Area of a PrismThe surface area S of a right prism is S = L + 2B (or S = Ph + 2B), where L is its lateral area and B is the area of a base.

Mar 31­11:35 AM

Find the lateral area and surface area of the cylinder. Round to the nearest tenth.

L = 848.2 mm2

S=1201.6mm2Answers

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Objective 5: Volume of Prisms and Cylinders

Read the bottom of p. 571 Beckmann hardcover, "The Volume Formula for Prisms and Cylinders."

Class Activities: 13N and 13O

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Volume of Prisms & Cylinders

Volume of a Prism

V = Bh

where B is the area of the base, and h is the height of the prism.

Volume of a Cylinder

V = Bh or V = πr2h

where B is the area of the base (πr2), and h is the height of the prism.

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V=340mm3 V=540ft3AnswersAnswers

Find the volume of each prism or cylinder.

V=981.0cm3Answers

12.49 cm

12 ft

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Cavalieri's Principle: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume.

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Find the volume.

V=110.72ft3Answers

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Beckmann 13NWhy the Volume Formula for Prisms and Cylinders Makes Sense

1. What does it mean to say that a solid shape has a volume of 12 cubic inches?

2. Explain why the (height) X (area of base)

formula gives the correct volume for right prisms and cylinders. Do so by imagining that the prism or cylinder is built with 1-unit-by-1-unit-by-1-unit clay cubes (or other cubes that could be cut into pieces, if necessary). Think about building the shape in layers.

~How many layers would you need?~How many cubes would be in each layer?

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Build prisms on the bases shown below. Use these models to help you explain why the volume formula is valid.

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Cavalier's Principle:

http://www.jimloy.com/cindy/cavalier.htm

4. Use the result of part 2 and Cavalieri's principle to explain why

(height) X (area of base)

gives the correct volume for an oblique prism or cylinder. explain why the height should be measured perpendicular to the bases, and not "on the slant."

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Chart with volume formulas (as needed)

http://www.mathwords.com/v/volume.htm

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Now practice with volume:

http://www.aaastudy.com/geo.htm

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Objective 6: Pyramids and Cones

Include the oblique cone and pyramid in your work.

(Beckmann hardcover 555-557)

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Surface Areas of Pyramids and Cones

Note: The lateral faces of a pyramid intersect at a common point called the vertex. Two lateral faces intersect at a lateral edge. A lateral face and the base intersect at a base edge. The altitude is the segment from the vertex perpendicular to the base.

A regular pyramid has a base that is a regular polygon and the altitude has an endpoint at the center of the base. All the lateral edges are congruent and all the lateral faces are congruent isosceles triangles. The height of each lateral face is called the slant height l of the pyramid.

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~Lateral Area of a Regular PyramidL = (1/2)Pl

where l is the slant height and P is the perimeter of the base.

~Surface Area of a Regular PyramidS = (1/2)Pl + B

where l is the slant height, P is the perimeter of the base, and B is the area of the base.

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L= 260 m2 S = 438.3 in2

Find the surface area with a square base.

AnswersAnswers

Find the lateral area with a square base.

Find the surface area with a regular pentagon base.

S = 84.78 in2Answers

12.65 in.

7.13 in.slant height

B ≈ 13.48

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Surface Area of ConesS = πrl + πr2

where r is the radius of the base and l is the slant height.

Lateral Area of ConesL = πrl

where r is the radius of the base and l is the slant height.

If the axis of a cone is also the altitude, then the cone is a right cone. If the axis is not the altitude, then the cone is an oblique cone.

Apr 5­11:09 AM

Find the lateral area. Find the surface area.

L = 1.76π≈5.5mm2Answers L = 286.9in2Answers

9.22 in. 

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February 08, 2012

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Cone Surface Area Activity:Beckmann Activity book p. 317

Class Activity 13J: #3Make a pattern for a cone such that the base is a circle of radius 2 inches and the cone without the base is made from a half-circle. Determine the total surface area of your cone. Explain your reasoning.

#4Make a pattern for a cone such that the base is a circle of radius 2 inches and the lateral portion of the cone is made form part of a circle of radius 6 inches. What fraction of the 6-inch circle will you need to use? Determine the total surface area of your cone. Explain your reasoning.

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Volumes of Pyramids & Cones

Volume of a Pyramid

V = (1/3)Bh

where B is the area of the base, and h is the height of the pyramid.

Volume of a Cone

V = (1/3)Bh or V = (1/3)πr2h

where B is the area of the base (πr2), and h is the height of the cone.

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V=392.7cm3Answers

Find the volume of each pyramid or cone.

V=210ft3Answers

2.89 cm

5.77 cm

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V=1005.3ft3Answers V=36.7cm3Answers

Find the volume of each pyramid or cone.

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Volume of Pyramid Activities:Compare the volume of a pyramid with the volume of a rectangular prism (13P p. 322)

1. Cut out, fold, and tape the patterns on pp. 447 and 447A to make an open rectangular prism and an open pyramid with a square base.

2. Verify that the prism and the pyramid have bases of the same area and have equal heights.

3. Just by looking at your shapes, make a guess: How do you think the volume of the pyramid compares with the volume of the prism?

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4. Now fill the pyramid with beans, and pour the beans into the prism. Keep filling and pouring until the prism is full. Based on your results, fill in the blanks in the equations that follow:

volume of prism = __ X volume of pyramid

volume of pyramid = __ X volume of prism

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The 1/3 in the volume formula for pyramids and cones:Class activity 13Q Beckmann pp. 322-323

According to the volume formula, a right pyramid that is 1 unit high and has a 1 -unit-by- 1 -unit square base has volume

1/3 X 1 X (1 X 1) unit3 = 1/3 unit3 (Troy - need to fix the unit "3"s. I don't know how to do a power.)

Now pretend that you don't yet know the volume formula for pyramids and cones. This class activity will help you use principles about volumes to explain where the 1/3 comes from.

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1. Cut out three of the four patterns on pages 449-451. (The fourth is a spare.) Fold these patterns along the undashed line segments, and glue or tape them to make three oblique pyramids. Make sure the dashed lines appear on the outside of each oblique pyramid.

2. Fit the three oblique pyramids together to make a familiar shape. What shape is it? What is the volume of the shape formed from the three oblique pyramids? Therefore, what is the volume of one of the oblique pyramids?

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Volume versus Surface Area and HeightClass Activity 13T - Beckmann p. 325

1. Young children sometimes think that taller containers necessarily hold more than shorter ones. Make or describe two open-top boxes such that the taller box has a smaller volume than the shorter box.

2. Students sometimes get confused between the volume and surface area of a solid shape and about how to calculate volume and surface area.

a. Discuss the distinction between volume and surface area.b. Discuss the differences and similarities in the way we

calculate the volume and surface area of a rectangular prism.

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HOMEWORK:

Beckmann hardcover pp. 568 #2a, 2b, and 5

Reading: Beckmann hardback pp. 545-546 (Pythagorean Theorem)

Homework assignment: Worksheet