LESSON 10.2 Cylinders Expressions, - 8th Grade...
Transcript of LESSON 10.2 Cylinders Expressions, - 8th Grade...
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Oatmeal
r r
ℓ = 2πr
h h
C = 2πr
Oatmeal
ESSENTIAL QUESTION
EXPLORE ACTIVITY
How do you find the surface area of a cylinder?
Modeling the Surface Area of a CylinderJust as you did with a prism, you can use a net to help
you find a formula for the surface area of a cylinder.
The lateral area of a cylinder is the area of the curved
surface that connects the two bases. The net shows
that the lateral surface is a rectangle.
Use the diagram of the oatmeal container and its net.
Imagine unrolling the container’s lateral surface to form a rectangle.
What dimension of the cylinder matches the rectangle’s length?
What dimension of the cylinder matches the rectangle’s height?
Express the lateral area of a cylinder in words and as a formula using
the given variables.
Lateral area = Area of one = Length · Height
L = (C = circumference, h = height)
L = (r = radius, h = height)
Express the total surface area of a cylinder in words and as a formula
using the given variables.
Total surface area = Area of two + Area of one
S = ( B = base area, C = circumference, h = height)
S = (r = radius, h = height)
Reflect1. Communicate Mathematical Ideas How is the process for finding
lateral and total surface area of a cylinder like the process for a prism?
A
B
C
L E S SON
10.2Surface Area of Cylinders
8.7.B
Expressions, equations, and relationships—8.7.B … make connections to the formulas for lateral and total surface area and determine solutions for problems involving … cylinders.
The formula for the area of a circle is A = π r 2 .
The formula for the circumference of a circle is C = 2πr.
275Lesson 10.2
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Finding the Surface Area of a CylinderThe total surface area of a cylinder is the area of the bases plus the lateral area.
Find the lateral and total surface area of the cylinder
to the nearest tenth. Use 3.14 for π.
Find the lateral area.
L = 2πrh
≈ 2(3.14)(4)(7)
≈ 175.84
The lateral area is about 175.8 square feet.
Find the total surface area.
S = 2π r 2 + L
≈ 2(3.14)( 4 2 ) + 175.84
≈ 100.48 + 175.84
≈ 276.32
The total surface area of the cylinder is about 276.3 square feet.
EXAMPLE 1
STEP 1
STEP 2
Find the lateral and total surface area of each cylinder. Round your
answers to the nearest tenth. Use 3.14 for π.
2. L:
S:
3. L:
S:
YOUR TURN
8.7.B
RadiusBase
Height
Lateral and Total Surface Area of a Cylinder
The lateral area L of a cylinder with
height h and radius r is the circumference
of the base times the height.
L = Ch or L = 2πrh
The total surface area S of a cylinder
with height h and radius r is twice the
area of a base B plus the lateral area L.
S = 2B + L or S = 2π r 2 + 2πrh
Use the formula for the lateral area of a cylinder.
Use the formula for the totalsurface area of a cylinder.
Add.
Substitute.
Simplify.
Simplify.
Substitute.
Unit 3276
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Finding the Surface Area of a Cylinder in a Real-World SituationYou can find many examples of cylinders in your kitchen.
The can shown is 11 inches high and has a
diameter of 9 inches. How many square inches
of paper are needed for the label? How many square
inches of metal are needed to make the entire can?
Round your answers to the nearest whole number.
Use 3.14 for π.
Find the radius of the base.
d = 9 in., so r = 4.5 in.
To find the area of the label, find the lateral area.
L = 2πrh
≈ 2(3.14)(4.5)(11)
≈ 310.86
It takes about 311 square inches of paper to make the label.
To find the amount of metal, find the total surface area.
S = 2π r 2 + 2πrh
≈ 2(3.14)( 4.5 2 ) + 2(3.14)(4.5)(11)
≈ 127.17 + 310.86
≈ 438.03
It takes about 438 square inches of metal to make the can.
EXAMPLEXAMPLE 2
STEP 1
STEP 2
STEP 3
4. How many square inches of cardboard are needed for
the lateral area of the raisin container shown? What
is the total surface area of the container? Round your
answers to the nearest tenth. Use 3.14 for π.
YOUR TURN
8.7.B
The radius is half
Use the formula for the lateral area of a cylinder.Substitute.
Multiply.
Use the formula for the total surface area of a cylinder.Substitute.
Simplify.
Add.
of the diameter.
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Guided Practice
1. Draw a net of the cylinder. Label the radius and the height. Then
find the lateral area and total surface area. Round your
answers to the nearest tenth. Use 3.14 for π. (Explore Activity and Example 1)
2. A can of tuna fish has a height of 1 inch and a diameter of 3 inches.
How many square inches of paper are needed for the label? How
many square inches of metal are needed to make the can including
the top and bottom? Round your answers to the nearest whole
number. Use 3.14 for π. (Example 2)
It takes about square inches of paper to make the label.
It takes about square inches of metal to make the can.
3. How do you find the total surface area of a cylinder?
ESSENTIAL QUESTION CHECK-IN??
Find the circumference of the circle. Use 3.14 for π.
C = 2πr ≈ 2 · 3.14 · ≈ inches
The lateral area is the circumference times the
height of the can.
L ≈ 31.4 · ≈ in2
Find the area of the two bases. Use 3.14 for π.
2B = 2π r 2 ≈ 2(3.14) ( ) ≈ in2
Add the area of the bases and the lateral area.
S ≈ + ≈ in2
Unit 3278
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Independent Practice10.2
Find the lateral and total surface area of each cylinder. Round your answers
to the nearest tenth. Use 3.14 for π.
4. 5. 6.
7. A container of bread
crumbs has a radius of
4 inches and a height of
8 inches.
8. A rain barrel has a
diameter of 18 inches
and a height of 3 feet.
9. A drum has a diameter of
14 inches and a height of
5.5 inches.
10. Multistep A pipe is 25 inches long and has a diameter of
5 inches. What is the lateral area of the pipe to the nearest
tenth? Use 3.14 for π.
11. Multistep A size D battery has a diameter of 32 millimeters
and a length of 5.6 centimeters. What is the lateral area of the
battery to the nearest square centimeter? Use 3.14 for π.
12. What If? Carol is designing an oatmeal container. Her first
design is a rectangular prism with a height of 12 inches, a
width of 8 inches, and a depth of 3 inches.
a. What is the total surface area of the container to the
nearest square inch? Use 3.14 for π.
b. Carol wants to redesign the package as a cylinder with
the same total surface area as the prism in part a. If the
radius of the cylinder is 2 inches, what is the height of
the cylinder? Round your answer to the nearest inch.
Use 3.14 for π.
13. Vocabulary How do the lateral and total surface area of a cylinder differ?
8.7.B
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14. Multiple Representations The formula for the total surface area of a
cylinder is S = 2π r 2 + 2πrh. Explain how you could use the Distributive
Property to write the formula another way.
15. Persevere in Problem Solving The
treasure chest shown is a composite figure.
What is the surface area of the treasure
chest to the nearest square foot? Explain
how you found your answer.
16. Analyze Relationships A square prism has the same
height as the cylinder shown. The perimeter of the prism’s
base equals the circumference of the cylinder’s base.
a. Find the side length of the prism’s base to the
nearest centimeter.
b. Which figure has the greater lateral area? Which has the greater total
surface area? Which has the greater volume? Explain.
17. Communicate Mathematical Ideas A cylinder has a circumference of
16π inches and its height is half the radius of the cylinder. What is the
total surface area of the cylinder? Give your answer in terms of π. Explain
how you found your answer.
FOCUS ON HIGHER ORDER THINKING
Unit 3280
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