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Geometry

Week 32: April 13-17, 2015

G.13

The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems.

G.14

The student will use similar geometric objects in two- or three-dimensions to

a) compare ratios between side lengths, perimeters, areas, and volumes;

b) determine how changes in one or more dimensions of an object affect area and/or volume of the object;

c) determine how changes in area and/or volume of an object affect one or more dimensions of the object; and

d) solve real-world problems about similar geometric objects.

Monday: Finish Quiz and Spheres

Homework: SOL G.13 Study Guide problems (1-6)

Tuesday: Volume and Surface Area of Spheres

Homework: Page 868 (1-9) odd

Wednesday: Activities 1 and 2: Investigating Similar Solids

Homework: Geometry Online

Thursday: Review for Unit Test

Homework: Study (see study guide on what will be tested)

Friday: Unit Test

Follow your passion, be prepared to work hard and sacrifice, and, above all, don't let anyone limit your dreams.

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Geometry Study Guide Mrs. Grieser

-1-

Name: _____________________________ Date: _______________ Block: _________

SOL G.13 Surface Area and Volume Study Guide

Know how to:

Find the surface area and volume of solids: prisms, cylinders, pyramids, cones, and spheres

Use the SOL formula sheet to help. The formula sheet may be used on this quiz, but you must

bring your own copy.

Use area and volume ratios to solve problems: the ratio of perimeters is a:b, the ratio of

areas is a2:b2, and the area of volumes is a3:b3.

Study questions – round to the nearest hundredth if necessary:

1) Find the lateral area, total surface area, and volume of the prism.

2) Find the lateral area, total surface area, and volume of the cylinder.

3) Find the lateral area, total surface area, and volume of the cone.

4) Find the lateral area, total surface are, and volume of the regular pyramid.

5) Find the surface area and volume of the solid.

6) A cone with radius 6cm and height 12cm is filled to capacity with liquid. Find the height of a cylinder with radius 4 cm that will hold the same amount of liquid.

7) If the diameter of a soap bubble is known to be 2 mm, what is the surface area of the bubble? What is the volume?

8) A pipe is 300 cm long and has inside radius 4 cm and outside radius 5 cm. Find the volume of metal. Hint: draw a diagram.

9) The volume of a sphere is

36π. Find the surface area.

Geometry Study Guide Mrs. Grieser

-2-

10) A pharmacist is filling medicine capsules. The capsules are cylinders with half spheres on each end. If the length of the cylinder is 12 mm and the radius is 2 mm, how many cubic mm of medication can one capsule hold?

11) The number of square centimeters in the surface area of a sphere is twice the number of cubic centimeter in the volume of a sphere. Find the radius of the sphere.

12) A cylinder has a surface area of 1005 in2. Find the height of the cylinder if the radius is 8 in (round to the nearest whole number).

13) A regular square pyramid is inscribed in a cone with radius 4 cm and height 4 cm. (a) What is the volume of the pyramid? (b) Find the slant height of the cone and the pyramid.

14) Are the figures below similar? Explain.

15) Cube C has a surface area of 36 units2 and Cube D has a surface area of 144 units2. Find the scale factor of C to D.

16) Two rectangular prisms are similar. Their scale factor is 3:5. For the smaller prism, the surface area is 90 cm2 and the volume is 54 cm3. Find the surface area and volume of the larger figure.

17) Cone A is similar to cone B. The scale factor of A:B is 2:3. The surface area and volume of cone A are 90π m2 and

100π m3. Find the surface

area and volume of cone B (express answers in terms of π).

18) The volume of sphere A is 343π in3 and the volume of

sphere B is 8π in3. Find

the scale factor of A:B.

19) At the MetroPlex movie theater, popcorn is served in a box. At the CinemaPlex movie theater, popcorn is served in a cylindrical container. At home, Mom serves popcorn in a bowl (hemisphere in shape). Based upon the given dimensions, where are you getting the most popcorn? (Disregard the thickness of the container.)

Mathematics Enhanced Scope and Sequence – Geometry

Virginia Department of Education © 2011 3

Activity Sheet 1: Investigating Similar Solids Name Date

1. On the graph paper, draw a rectangle (not a square) with length and width between 2 and 5 units.

2. Leaving at least three rows between the two rectangles, draw another rectangle with length and width that are twice the length and width of the original rectangle.

3. Leaving at least three rows between the rectangle and the other rectangles, draw a third rectangle with length and width that are three times the length and width of the ORIGINAL

rectangle.

4. Complete the table below.

Small Rectangle Medium Rectangle Large Rectangle

Length Width

Perimeter Area

5. How many of the small rectangles fit in the medium rectangle? _______

6. How many of the small rectangles fit in the large rectangle? _______

7. Compute the ratios of the three rectangles’ lengths, perimeters , and areas using the data above. Complete the table below. Be sure to reduce all ratios!

Ratios: Small:Medium Small:Large Medium:Large

Length

Perimeter Area

8. How are the ratios of the areas related to your answers to #2 and #3?

9. Use cubes to build a small rectangular prism with length and width the same as the small rectangle and with height 1. How many rectangles do you need? ______

Sketch this prism on your small rectangle by adding diagonals as shown in figure 1.

10. If you have enough cubes, build a medium rectangular prism with length and width the same

as the medium rectangle and with height 2. How many rectangles do you need? ______ Sketch this prism on your medium rectangle by adding diagonals as shown in figure 1, only two deep.

11. If you have enough cubes, build a large rectangular prism with length and width the same as

the large rectangle and with height 3. How many rectangles do you need? ______ Sketch this prism on your large rectangle by adding diagonals as shown in figure 1, only three deep.



12. Complete the rest of the table below.

Small Rectangular Prism

Medium Rectangular Prism

Large Rectangular Prism

Height 1 2 3

Surface Area

Volume

13. Compare the volumes of the prisms to your answers to #7, #8, and #9. Explain.

14. Similar solids are solids of the same type (like spheres and rectangular prisms) that have proportional linear measures (like length, width, height, perimeter, and radius). Are your three rectangular prisms similar? How do you know?

15. How many of the small rectangular prisms fit in the medium rectangular prism? _______

16. How many of the small rectangular prisms fit in the large rectangular prism? _______

17. Compute the ratios of the three rectangular prisms’ heights, surface areas , and volumes using the data above. Complete the table below. Be sure to reduce all ratios!

Ratios: Small:Medium Small:Large Medium:Large Height

Surface Area

Volume

18. How are the ratios of the volumes related to your answers to #14 and #15?

19. What is the scale factor of the small rectangular prism to the medium rectangular prism? ___________

20. What is the scale factor of the small rectangular prism to the large rectangular prism? ___________

21. How are the scale factors (#16) related to the corresponding ratios of surface areas and volumes?



Activity Sheet 2: Investigating Similar Solids Name Date

You must show your work including proportions or other equations where appropriate (# 4–8). Do

not round your answers. (i.e., π31125 cm3, rather than 6121.6 cm3)

The ratio of the heights of two similar cones is 2:5. The radius of the smaller cone is 6 cm.

1. What is the scale factor of the two cones?

2. What is the radius of the larger cone?

3. What is the ratio of the lateral areas?

4. The lateral area of the smaller cone is 72π square centimeters. What is the lateral area of the larger cone? (Hint: Use #3)

5. What is the ratio of the volumes?

6. The volume of the larger cone is π31125 cubic centimeters. What is the volume of the

smaller cone? (Hint: Use #5)

7. Find the slant heights and heights of the two cones. (You may already have found one or more of these.)

8. Find the angle x formed by a radius and the sides of the cones.

9. Complete the table below.

r L.A. V l h

Small Cone 72π Large Cone π31125

Ratio 2:5

10. Compute the volumes of the following solids: a) A cylinder with height r and radius r b) A hemisphere with radius r c) A cone with height r and radius r

11. Draw diagrams of the three figures. Try to draw them so they have the same radius and height. Compare the volumes. How many cones of water would it take to fill the cylinder? The hemisphere?



Activity Sheet 4: Exploring Volume Name Date

1. Which will carry the most water in a given length—two pipes with one having a 3 dm radius and the other a 4 dm radius, or one pipe with a 5 dm radius? Explain.

2. A company delivers 36 cartons of paper to your school. Each carton measures 40 cm x 30 cm x 25 cm. Is it possible to fit all cartons in an empty storage closet 1 m x 1 m x 2 m? Justify your

conclusion with a visual explanation.

3. You have studied the pyramids and want to make a scale model of a pyramid with a square base and sides that are isosceles triangles. How much clay is required if the base of the actual

pyramid is 30 m on each side and the height of the pyramid is 30 m? Your scale is 1 cm = 15 m.

4. A movie theater decides to change the shape of its popcorn

holder from a rectangular box to a pyramidal box. The tops of both boxes are the same and the height remains the same. If the rectangular bag of popcorn cost $4.00, what is a fair price for the new box?

5. A manufacturer of globes that are approximately 1 m in diameter packs the globes in 1-cubic-meter boxes for shipping. How much packing material (foam peanuts) is needed for a shipment

of 100 globes?

6. Take two sheets of paper the same size. Roll one sheet vertically and tape to form a right circular cylinder. Roll the second sheet horizontally, and tape it to form a second right circular

cylinder. Tape each cylinder so that there is no overlap of paper—i.e., the edges should meet exactly. If each cylinder were filled with popcorn, would they contain the same amount?

Explain and justify your answer.

Chapter 9: Surface Area/ Volume name ________________________Lesson 9-5: Similar Objects date ______________Classwork period ___

Suppose the two right cylinders shown are similar. find:

________1. The scale factor.

________2. Write a proportion and solve for h.

________3. the ratio of base areas.

________4. the ratio of lateral areas.

________5. the ratio of volumes.

Complete the table about similar solids.

1. 2. 3. 4. 5. 6. 7. 8.

81:1

1:21664:27

9:25

5:81:52:33:4

ratio of volumes

ratio of areas

ratio of corresponding segments

=12

B = 8

= h

B = 10

Questions 9- 11 refer to two similar prisms with a scale factor of 3:7.

_______9. The shortest edge of the larger prism is 21 cm long. How long is the shortest edge of the smaller prism?

_______10. The base area of the larger prism is 98 cm2 . Find the base area of the smaller prism.

_______11. The volume of the smaller prism is 135 cm3 . Find the volume of the larger prism.

_______12. The total areas of two similar cones have the ratio 25:9. Find the ratio of their volumes.

The two right rectangular prisms below are similar.

________13. Find the ratio of the surface areas.

________14. Suppose the volume of the smaller prism is 8 cubic meters. Find the volume of the larger prism.

________15. True/False: If two solids are congruent, then their volumes are equal.

5 cm2 cm

Chapter 9: Surface Area/ Volume name ________________________Lesson 9-5: Similar Objects date ______________Homework period ___

Complete the table about similar solids. 1. 2. 3. 4. 5. 6. 7.

121:49729:1000

25:9216:27

9:1

5:22:3

ratio of volumesratio of areas

ratio of corresponding segments

_________8. Two cones have radii 6 and 8. The heights are 30 and 40, respectively. Are the cones similar?

_________9. Two cylinders have radii 6 and 15. The heights are 50 and 125, respectively. Are the cylinders similar?

_________10. The heights of two right prisms are 9 and 15. The bases are squares with sides 27 and 45, respectively. Are the prisms similar?

_________11. Two cones have radii 14 and 42. The heights are 24 and 96, respectively. Are the cones similar?

_________12. Two rectangular prisms are similar. The measures of two corresponding sides are 1 m and 2 m. What is the ratio of the volumes of the prisms?

_________13. Two soccer balls are made with the same material have a radii of 8 cm and 15 cm. If the larger ball weighs 45 kg, how much does the smaller ball weigh?

_________14. The area of a triangle is 48 cm. If the base of this triangle is 12 cm, what is the length of the altitude to this base?

_________15. The area of square ABCD is 81 square centimeters. What is the number of centimeters in the perimeter of the square?

Geometry Online! Name __________________________________________

PRACTICE – Similar Geometric Objects - G.14 Date _________Period _____

3



30 cm

40 cm

10 cm

Activity Sheet 1: Finding Formulas

1. After you purchase a gift for a friend, you decide to cover the sides and bottom of the gift box with wrapping paper. A diagram of the box with its dimensions appears below.

a) How much wrapping paper will you need to cover the sides and bottom of the box? b) Your gift box is called an open box because it has no top surface.

If this were a closed box with a top surface, how much additional paper would be required to cover the top surface? How much total paper would be required?

c) How can you generalize the process you used to find the surface area of the closed box? d) Let l = length, w = width, and h = height of the box. e) Compare the formula you and your partner developed to that of another group. Did you

have the same result? You should be able to justify your formula to your classmates. 2. If your gift were a can of tennis balls, the surface area would be the surface of the cylinder

(the lateral area) plus the areas of the top and bottom (the bases). Use a can (soup can,

soda can, tennis ball can) for this activity. a) Wrap a piece of paper around the can, trim it to fit exactly, and spread it out

flat. What shape is it? How can you find its area? What relationship does the length of the label have to the can? The height of the label?

b) What shape are the bases of the can? Are the two bases congruent? What is the area of each base?

c) The surface area of the can = the lateral area + the area of the two bases. For your can, what is the surface area? Use your calculator to find decimal approximations to the nearest tenth.

3. The surface area of a sphere is more difficult to figure out. On a globe, a great circle is a circle

drawn so that when the sphere is cut along the line, the cut pass es through the center of the sphere. The equator is a great circle on a globe.

a) Draw a great circle on an orange, and carefully cut the orange in half along the line of the great circle. Trace five cut halves on a piece

of waxed paper. b) Carefully peel both halves of the orange, and fill in as many

circles as you can with the peel. How many circles did your

group fill? How does this compare with the findings of other groups? What is the class estimate for the number of great

circles that can be filled by the peel? c) Using one of your great circle tracings, find the radius of your orange

and the area of one great circle. d) Given the area of one great circle and your estimate of the number of circles that can be

filled by the peel, what is the surface area of the orange? e) What is the general formula for the surface area of a sphere in terms of its radius?



Scale: 1 cm = 15 m

30 m

30 m

30 m

Activity Sheet 2: Making Nets Name Date

1. A net is a flattened paper model of a solid shape. For example, the net shown to the right, when folded, makes a cube. Can you draw a different net which, when folded, will also make a cube? If so, draw it, cut it out, and fold it to test your drawing.

2. A net is helpful because it represents the surface area of a shape. Take a box and cut it into a net. Note whether your box is open or closed. Sketch your box and its net.

Use the formula you derived in “Finding Formulas” problem 1 to find the surface area of your box. Explain to a classmate how your net relates to your formula.

3. Now sketch a net of the can you used in “Finding Formulas” problem 2. How does this net relate to the surface area formula you found?

4. Sketch a net of the pyramid shown to the right. Use your net to find the

surface area of the pyramid.



Activity Sheet 3: Solving Problems Name Date

1. Two cylindrical lampshades 40 centimeters in diameter and 40 centimeters high are to be covered with new fabric. The fabric chosen is 1 meter wide. If you purchase a 1.5-meter length of this fabric, will you have enough to cover both lampshades? Justify your answer.

2. An umbrella designer has created a new model for an umbrella that, when opened, has the

form of a hemisphere with a diameter of 1 meter. If a dozen sample models are to be made using a special waterproof material, approximately how much waterproof fabric will be

needed, allowing 0.5 meter for seams and waste for each model? Explain your plan,

strategies, and how you solved the problem.

UNIT 9 TEST (3-D FIGURES) G.13

The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems.

G.14

The student will use similar geometric objects in two- or three-dimensions to

a) compare ratios between side lengths, perimeters, areas, and volumes;

b) determine how changes in one or more dimensions of an object affect area and/or volume of the object;

c) determine how changes in area and/or volume of an object affect one or more dimensions of the object; and

d) solve real-world problems about similar geometric objects.

Surface Area

Cylinders

Prisms

Pyramids

Spheres

Cones

Volume

Prisms (including cubes)

Cones

Pyramids

Spheres

Complex 3-D shapes

Comparing ratios between side lengths, perimeters, areas, and volumes. What happens if you change

one or more attributes.

Similar figures (Using proportions)

Finding surface area when given volume.

Area and Perimeter of Squares

You still need to know the Pythagorean Theorem.

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