GEOMETRY AND MEASUREMENT Student Pages for Packet 3 ... GEO3-SP.p… · GEO3.2 Cylinders Derive the...

Name ___________________________ Period __________ Date ___________

Geometry and Measurement Unit (Student Packet) GEO3 – SP

GEO3.1 Rectangular Prisms

Visualize and create two-dimensional nets for three-dimensional figures.

Find the surface area and volume of rectangular prisms. Derive and use formulas for the surface area and volume of

rectangular prisms. Use appropriate units of measurement for length, area, and

volume. Write expressions in different forms using properties of arithmetic.

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GEO3.2 Cylinders Derive the formulas for surface area and volume of cylinders. Compute the surface area and volume of cylinders.

9

GEO3.3 Investigations with Solids

Explore the effect of a scale factors on length, surface area, and volume of a prism.

Compare the volumes of cylinders that have the same lateral area.

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GEO3.4 Vocabulary, Skill Builders, and Review 22

GEO3 STUDENT PAGES

GEOMETRY AND MEASUREMENT

Student Pages for Packet 3: Surface Area and Volume

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Surface Area and Volume

Geometry and Measurement Unit (Student Pages) GEO3 – SP0

WORD BANK (GEO3)

Word Definition Example or Picture

net

surface area

volume

rectangular prism

cylinder

lateral area

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Surface Area and Volume 3.1 Rectangular Prisms


RECTANGULAR PRISMS

Ready (Summary)

We will compute the surface area and volume for rectangular solids.

Set (Goals)

Visualize and create two-dimensional nets for three-dimensional figures.

Find the surface area and volume of rectangular prisms.

Derive and use formulas for the surface area and volume of rectangular prisms.

Use appropriate units of measurement for length, area, and volume.

Write expressions in different forms using properties of arithmetic.

Go (Warmup)

Give an example of the: 1. associative property of addition (assoc )__________________________________ 2. commutative property of multiplication (comm )____________________________ 3. identity property of multiplication (identity )___________________________________

4. distributive property of multiplication over addition (dist)_______________________ 5. commutative property of addition (comm )________________________________ 6. additive inverse property (inv )__________________________________________ 7. identity property for addition (identity )____________________________________ 8. associative property of multiplication (assoc )_________________________________ Sam

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FIXED VOLUME Use 12 cubes to make different rectangular prisms. Use the grid paper on the next page to draw some nets for the prisms. Record the length, width, height, and the area of the base in the table. Find the volume and surface area for each rectangular prism.

Length (L)

Width (w)

Height (h)

Area of Base (B)

Volume (V)

Surface Area (S)

1. Explain in words how you found the surface area of each prism. ________________ __________________________________________________________________ 2. What are the dimensions of the prism with the greatest surface area? ___________ 3. What are the dimensions of the prism with the least surface area? ______________

Use L, w, h Use B, h

Surface Area of rectangular

prism S =

Volume of rectangular

prism V = V = Sam

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MAKING NETS

Use 12 cubes to make different solid rectangular prisms. Draw some nets for the rectangular prisms.

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FINDING SURFACE AREA AND VOLUME

A rectangular prism has a length of 3.5 cm, a width of 4.5 cm, and a height of 5 cm.

1. Sketch the rectangular prism and label its dimensions.

2. Find the area of its base.

Formula: _______________________ Substitute: Answer: ________________________

3. Find its volume.

Formula: _______________________ Substitute: Answer: ________________________

4. Find its surface area.

Formula: _______________________ Substitute: Answer: ________________________ Sam

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SURFACE AREA FORMULAS Write a formula to match each student’s thinking.

1. Raoul said, “I found the surface area of the rectangular solid by adding up the area of each flat surface.”

2. Sanjay said, “I found the area of the base and doubled it, then I found the area of the front

and doubled it, and finally I found the area of one of the sides and doubled it. Then I added it all together.”

3. Rebecca said, “There are 2 of each type of face, so I can find the area of each type of face, add them together, and then double the sum.”

4. Allyse said, “I found the perimeter of the base of the prism and multiplied it by the height to

get the surface area of all the vertical sides (called the “lateral area”). Then, I added the area of the two bases to that total.”

State the property of arithmetic or equality that explains why each of these statements is true: (Let L = length, w = width, and h = height.)

Statement Property

5. Lw wL

6. 2 2 2 2( )Lw wh Lh Lw wh Lh

7. Lw Lw 1Lw 1Lw

8. 1 1 (1 1)Lw Lw Lw

9. Lw wh whLw

10. ( ) ( )Lw wh Lh Lw wh Lh Sample

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USING SURFACE AREA AND VOLUME FORMULAS 1

The length of a rectangular prism is 10 cm. The height is half the length. The width is 3 less than the length.

1. Determine the dimensions of the rectangular prism.

length: L = ____________________

width: w = ____________________

height: h = ____________________


3. Find the surface area.

Formula: S = 2(Lw whLh)

Substitute:

Answer: ________________________

4. Find the volume.

Formula: V = Lw h

Substitute:

Answer: ________________________

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USING SURFACE AREA AND VOLUME FORMULAS 1 (continued)

The width of a rectangular prism is 3 12

feet.

The length is 3 times as long as the width. The height is 1 foot more than the width.


length: L = ____________________

width: w = ____________________

height: h = ____________________



Formula: S = 2(Lw whLh)

Substitute: Answer: ________________________

8. Find the volume.

Formula: V = Lw h

Substitute: Answer: ________________________

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MY PRISM Find a container from home that is a rectangular prism and complete the questions. Make a net to cover your prism container and bring the net to class. What is your prism used for? ______________________________________________

1. Write the dimensions of your prism.

length: L = ____________________

width: w = ____________________

height: h = ____________________

4. Find the surface area of your prism to the nearest square centimeter.

2. Sketch and label the dimensions of your prism.

3. Sketch a net of your prism.

5. Find the volume of your prism to the nearest cubic centimeter.

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Surface Area and Volume 3.2 Cylinders


CYLINDERS

Ready (Summary)

We will make sense of the formulas for the surface area and volume of cylinders and apply them to solve problems.

Set (Goals)

Derive the formulas for surface area and volume of cylinders.

Compute the surface area and volume of cylinders.

Go (Warmup)

1. The formula for finding the circumference of a circle is: ______________________ 2. The formula for finding the area of a circle is: ______________________________ A circle has a circumference of 37.68 inches. (Use 3.14 as needed)

3. Find the radius of the circle. 4. Find the area of the circle.

A rectangular prism measures 2 ft. 5 ft. 6 ft.

5. Find the surface area of the prism. 6. Find the volume of the prism.

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FORMULA FOR SURFACE AREA OF A CYLINDER

Your teacher will show you a model of a cylinder.

1. Draw and name the shape of each base. Label its dimensions.

2. Draw and name the shape of the lateral area (“curved surface”) when it is laid out flat. Label its dimensions.

3. Draw a net of the cylinder. Label its dimensions.

4. Write a formula for the surface area of a cylinder.

5. Find the surface area of a cylinder if the base has a radius of 3.5 cm, and the

height 4 cm. (Use

227

.)

6. Find the surface area of a cylinder if the base has a diameter of 3 inches, and the height is 6 inches. (Use 3.14 .)

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FORMULA FOR VOLUME OF A CYLINDER 1. What is the formula for the volume of a rectangular prism in terms of the length (L), width

(w), and height (h)? 2. What is the formula for the volume of a rectangular prism in terms of the area of the base

(B) and height (h)? To find the formula for the volume of a cylinder, multiply the area of the base by the height of the cylinder. 3. What is the formula for the area of the base of a cylinder? 4. What is the formula for the volume of a cylinder in terms of the area of the base?

5. Find the volume of a cylinder if the radius of the base is 3.5 cm, and the height is 4

cm. (Use

227

.)

6. Find the volume of a cylinder if the base has a diameter of 3 inches and the height is 6 inches. (Use 3.14 .)

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USING SURFACE AREA AND VOLUME FORMULAS 2

The diameter of a cylinder is 14 centimeters. The height is 3 less than double the radius

1. Determine the dimensions of the cylinder.

radius: L = ____________________

height: w = ____________________

2. Sketch the cylinder and label its dimensions.


Formula: S = 22 2r rh

Substitute:

Answer: ________________________

4. Find the volume.

Formula: V = 2r h

Substitute:

Answer: ________________________

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USING SURFACE AREA AND VOLUME FORMULAS 2 (continued)

The circumference of a cylinder is 22 inches. The lateral area of the cylinder is 22 square inches.


radius: L = ____________________

height: w = ____________________



Formula: S = 22 2r rh

Substitute:

Answer: ________________________

8. Find the volume.

Formula: V = 2r h

Substitute:

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Surface Area and Volume 3.3 Investigations with Solid


INVESTIGATIONS WITH SOLIDS

Ready (Summary)

We will find the surface area and volume for a juice box. Then we will multiply each linear dimension of the container by a scale factor and observe how it affects the surface area and volume. We will make different cylinders that have the same lateral area and compare their volumes.

Set (Goals)

Explore the effect of scale factors on length, surface area, and volume of a prism.

Compare the volumes of cylinders that have the same lateral area.

Go (Warmup)

1. Measure the dimensions of the juice box provided by your teacher to the nearest

centimeter. Then, find the class average of each measurement rounded to the nearest centimeter.

Juice Box Dimension Your Measurements (cm) Class Average (cm)

Length

Width

Height

2. Sketch a net of the juice box and label its dimensions.

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JUICE BOX: SURFACE AREA 1. Find the surface area of the juice box. Use the “class average” dimensions from the

previous page.

Formula:

Substitute:

Surface Area: ________________________cm2 2. Multiply the linear dimensions of the juice box by a scale factor. Find the new surface area.

Multiply original dimensions by a scale factor of:

New Length (cm)

New Width (cm)

New Height (cm)

New Surface Area

(cm2)

new

original

S

S

1 (original)

2

3

10

n

3. Explain what happens to the surface area of the juice box when the linear dimensions are

multiplied by a scale factor. ________________________________________________________________________ ________________________________________________________________________ Sam

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JUICE BOX: VOLUME 1. Find the volume of the juice box. Use the “class average” dimensions.

Formula:

Substitute: Volume: ________________________cm3

2. Multiply the linear dimensions of the juice box by a scale factor. Find the new volume.

Multiply original dimensions by a scale factor of:

New Length (cm)

New Width (cm)

New Height (cm)

New Volume (cm2)

new

original

V

V

1 (original)

2

3

10

n

3. Explain what happens to the volume of the juice box when the linear dimensions are

multiplied by a scale factor. ________________________________________________________________________ ________________________________________________________________________ Sam

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SCALE FACTOR PRACTICE 1

1. When the linear dimensions of a rectangle are each multiplied by 5, what happens to the area of the rectangle?

2. When the linear dimensions of a rectangle are each multiplied by n, what happens to the

area of the rectangle? 3. When the linear dimensions of a prism are multiplied by n, what happens to the surface

area of the prism? 4. When the length of the side of a cube is multiplied by 4, what happens to the volume? 5. When the linear dimensions of a prism are multiplied by n, what happens to its volume? 6. The area of a rectangle is 2 square feet. If each dimension of the rectangle is multiplied by

6, what will be the area of the new rectangle? Show how you got the answer in TWO different ways.

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PAPER CYLINDERS

Make models to investigate changes in the area of the base and the volume when paper cylinders are formed using a letter-size piece of paper. On your model, write the letter of the cylinder, the height, radius, area of base, and volume.

Paper Cylinder P:

Roll a letter-size piece of paper parallel to the 8 12

-inch sides to make a cylinder. Tape the

two 8 12

-inch edges together.

Paper Cylinder Q: Roll a letter-size piece of paper parallel to the 11-inch sides to make a cylinder. Tape the two 11-inch edges together.

Paper Cylinder R:

Cut a letter-size sheet of paper in half to make two strips that are each 4 14

X 11 inches.

Tape the strips to make a rectangle that is 22 inches long. Roll the strip parallel to the 22-inch side to make a cylinder. Tape the two 22-inch edges together.

Paper Cylinder S:


X 11 inches.

Tape the strips to make a rectangle that is 22 inches long. Roll the strip parallel to the 4 12

-

inch side to make a cylinder. Tape the two 4 12


Paper Cylinder T:


X 8 12

inches.

Tape the strips to make a rectangle that is 17 inches long. Roll the strip parallel to the 5 12

-

inch side to make a cylinder. Tape the two 5 12


Paper Cylinder U:


X 8 12

inches.

Tape the strips to make a rectangle that is 17 inches long. Roll the strip parallel to the 17-inch side to make a cylinder. Tape the two 17-inch edges together.

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PAPER CYLINDER P The lateral area of Paper Cylinder P is _____________________.

1. Sketch a net of the cylinder and label its dimensions.

The height is _______________________.

The circumference of the base is_______.

2. Sketch the cylinder and label its dimensions. Find the radius. Use 3 .

The radius is _______________________.

3. Find the area of the base of the cylinder. Use 3 .

The area of the base is _______________.

4. Find the volume of the cylinder. Use 3 . The volume is _____________________. Sam

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MY CYLINDER My cylinder is ____________________________________. The lateral area of my cylinder is _____________________.

1. Sketch a net of the cylinder and label its dimensions.

The height is _______________________.

The circumference of the base is_______.

2. Sketch the cylinder and label its dimensions. Find the radius. Use 3 .

The radius is _______________________.

3. Find the area of the base of the cylinder. Use 3 .

The area of the base is _______________.

4. Find the volume of the cylinder. Use 3 . The volume is _____________________.

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COMPARING CYLINDERS

Use class calculations to complete this table: lateral area height area of base volume

Paper Cylinder P

Paper Cylinder Q

Paper Cylinder R

Paper Cylinder S

Paper Cylinder T

Paper Cylinder U

1. Name the cylinders in order, from least to greatest, based on lateral area. 2. Name the cylinders in order from least to greatest, based on height. 3. Name the cylinders in order, from least to greatest, based on the area of the base. 4. Name the cylinders in order, from least to greatest, based on the volume.

5. What do you think will happen to the volume of any cylinder that has a fixed lateral area as

the size of the height gets smaller? 6. What do you think will happen to the volume of any cylinder that has a fixed lateral area as

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Surface Area and Volume 3.4 Vocabulary, Skill Builders, and Review


FOCUS ON VOCABULARY (GEO3)

Select the word(s) from the word bank that best complete each sentence. 1. A soup can is an example of a _______________, which is a three-dimensional figure with

two parallel circular bases connected by a curved surface. 2. A _______________ for a three-dimensional figure is a pattern in a plane that can be cut

and folded to form the surface of the figure. 3. A _______________ is a polyhedron in which two faces are congruent parallel polygons,

and the other faces are parallelograms. 4. A _______________ is a positive factor by which all the linear measurements of an object

are multiplied in order to create a proportional enlargement or reduction of the object. 5. A cube has six congruent faces. Its _______________ is the sum of the areas of these

square faces. 6. The _______________ of a rectangular prism is the product of its length, width, and height. 7. A prism has two bases that are ______________ and ______________ polygons. 8. The lateral area of a cylinder is a __________________.

Word Bank

net surface area volume

prism cylinder parallelogram

congruent scale factor parallel Sample

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SKILL BUILDER 1

Draw a picture, write an appropriate formula, and substitute to solve each problem.

1. The perimeter of a regular hexagon is 354 meters. Find the length of the sides of the hexagon.

2. The area of a triangle is 4.5 cm. The base is 6 cm. Find the height of the triangle.

a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question


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SKILL BUILDER 2

Draw a picture, write an appropriate formula, and substitute to solve each problem.

1. Find the circumference of a plate with a diameter of 8 inches.

2. The area of a circle is 78.5 square meters. Find the radius.



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SKILL BUILDER 3 Compute. Show your work.

1. 3 3

8

34

2. 25

16

3. 3 2

315

6

4. 25

38

5. 23

56

6. 3 1

2 2 1

4

Compute. Show your work.

7. 0.34510.76

8. 1.4 0.98

9. 15.123

0.11

10. 0.83 2

11.

0.4 0.12016 12.

45.630.03

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SKILL BUILDER 4

The length of a rectangular prism is 12 feet. The width is half the length. The height is 2 less than the length.


length: L = ____________________

width: w = ____________________

height: h = ____________________



Formula:

Substitute:

Answer: ________________________

4. Find the volume.

Formula:

Substitute:

Answer: ________________________

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SKILL BUILDER 5

The base of a cylinder has a radius of 8 inches. The height of the cylinder is 3 inches.


radius: L = ____________________

height: h = ____________________



Formula:

Substitute:

Answer: ________________________

4. Find the volume.

Formula:

Substitute:

Answer: ________________________

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TEST PREPARATION (GEO3) Show your work on a separate sheet of paper and choose the best answer. 1. What is the surface area of a 14 in. 6 in. 9 in. rectangular prism?

A. 756 in 3 B. 756 in.2 C. 528 in.3 D. 528 in.2 2. What is the volume of a 7 in. 3 in. 5 in. rectangular prism?

A. 105 in.3 B. 105 in.2 C. 142 in.3 D. 142 in. 2 3. What is the new volume of a rectangular prism with original dimensions of

4 in. 2 in. 3 in. when its linear dimensions are multiplied by a scale factor of 2?

A. 48 in.2 B. 48 in.3 C. 192 in.2 D. 192 in.3 4. What is the surface area of a cylinder that has a base with a diameter of 18 ft. and a

height of 13 ft.? Round to the nearest whole number.

A. 234 ft.2 B. 734 ft.2 C. 1,243 ft.2 D. 3,306.42 ft.2 5. What is the volume of a cylinder that has a base with a radius of 5 mm and a height of 4

mm?

A. 314 mm3 B. 314 mm2 C. 125.6 mm3 D. 125.6 mm2

6. What is the new volume of a cylinder with a base of diameter 4 mm and a height of 8

mm when the dimensions are multiplied by a scale factor of 4?

A. 32 mm3 B. 96 mm3 C. 144 mm3 D. 288 mm3 7. The area of a circle is 49 square ft. If the diameter is multiplied by 4, what is the area

of the new circle?

A. 784 sq ft B. 196 sq ft C. 28 sq ft D. 14 sq ft

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KNOWLEDGE CHECK (GEO3)

Show your work on a separate sheet of paper and write your answers on this page. 2.1 Rectangular Prisms The length of a rectangular prism is 9 cm. The height is one-third of the length. The width is 4 less than the length. 1. Sketch and label the rectangular prism. 2. Find the surface area and volume of the rectangular prism. 2.2 Cylinders A cylinder has a base with a diameter of 5 ft. and a height of 16 ft. 3. Sketch a net of the cylinder and label the dimensions. 4. Find the surface area of the cylinder. 2.3 Investigations with Solids 5. Explain what happens to the surface area of rectangular prism when the linear dimensions

are multiplied by a scale factor.

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Home-School Connection (GEO3)

Here are some questions to review with your young mathematician. 1. Find the surface area and volume of a rectangular prism that measures 4 in. 2 in. 5 in. 2. Find the new volume of a juice box with original dimensions of 3 ft. 2 ft. 5 ft. when it is

multiplied by a scale factor of 3. 3. A cylinder has a base with a diameter of 6.2 mm and a height of 4.5 mm. Find its surface

area. Parent (or Guardian) signature ____________________________

Selected California Mathematics Content Standards

MG 7.2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.

MG 7.2.2 Estimate and compute the area of more complex or irregular two-and three-dimensional figures by breaking the figures down into more basic geometric objects.

MG 7.2.3 Compute the length of the perimeter, the surface area of the faces, and the volume of a three-dimensional object built from rectangular solids. Understand that when the lengths of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor.

MR 7.2.2 Apply strategies and results from simpler problems to more complex problems.

FIRST PRINTING DO NOT DUPLICATE © 2009

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GEOMETRY AND MEASUREMENT Student Pages for Packet 3 ... GEO3-SP.p… · GEO3.2 Cylinders Derive the...

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