Honors Geometry Unit 8

Prisms and Cylinders Lesson 2

Lateral Area, Surface Area, and Volume

Lesson 10-2: 3-D Views of Solid Figures

2

Different Views

Perspective view of a cone

the side(or from any side view)

the top the bottom

Different angle views of a cone

Lesson 10-2: 3-D Views of Solid Figures

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Example: Different Views

* Note: The dark lines indicated a break in the surface.

Front Left Right Back Top

Lesson 10-2: 3-D Views of Solid Figures

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Sketch a rectangular solid 7 units long, 4 units wide, and 3 units high using Isometric dot paper .

Step 1: Draw the top of a solid 4 by 7 units.

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Sketches

Lesson 10-2: 3-D Views of Solid Figures

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Step 2: Draw segments 3 units down from each vertex (show hidden sides with dotted lines).

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Sketches - continued

Lesson 10-2: 3-D Views of Solid Figures

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Step 3: Connect the lower vertices. Shade the top of the figure for depth if desired. You have created a

corner view of the solid figure.

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Sketches - continued

Lesson 10-2: 3-D Views of Solid Figures

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Nets and Surface Area

Imagine cutting a cardboard box along its edges and laying it out flat. The resulting figure is called a net.

A net is very helpful in finding the surface area of a solid figure.

top

back

end

front

end

bottom

=

Lesson 10-2: 3-D Views of Solid Figures

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Let’s look at another net.

This is a triangular pyramid. Notice that all sides lay out to be triangles.

=

Lesson 10-2: 3-D Views of Solid Figures

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Find the surface area of the figure using a net.

1. First, imagine the figure represented as a net.2. Find the area of each face.3. Find the sum of all the individual areas.

610

6

336

33

10

66 6=

Surface area = (6 x 10) + (6 x 10) + (6 x 10) + ½(6)(33) + ½ (6)(33) = 60 + 60 + 60 + 93 + 93 = 180 + 183

• lateral face – not base

• lateral edge – intersections of lateral faces, all parallel and congruent

• base edge – intersection of lateral face and base

• Altitude - perpendicular segment between bases

• Height – length of the altitude

• lateral area – sum of areas of all lateral faces

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Prism

Lateral Area of a Prism LA = Ph Surface Area : SA = Ph + 2B

= [Lateral Area + 2 (area of the base)]

Volume of a Right Prism (V )= Bh(P = perimeter of the base, h = height of prism, B = base area)

h

h

h

Triangular Prism

Lateral Area of a Prism

Find the lateral area of the regular hexagonal prism.

The bases are regular hexagons. So the perimeter of one base is 6(5) or 30 centimeters.

Answer: The lateral area is 360 square centimeters.

Lateral area of a prism

P = 30, h = 12

Multiply.

A. 162 cm2

B. 216 cm2

C. 324 cm2

D. 432 cm2

Find the lateral area of the regular octagonal prism.

Surface Area of a Prism

Find the surface area of the rectangular prism.

Surface Area of a Prism

Answer: The surface area is 360 square centimeters.

Surface area of a prism

L = Ph

Substitution

Simplify.

A. 320 units2

B. 512 units2

C. 368 units2

D. 416 units2

Find the surface area of the triangular prism.

Volume of a Prism

Answer: The volume of the prism is 1500 cubic centimeters.

V Bh Volume of a prism

1500 Simplify.

Find the volume of the prism.

A. 6480 in3

B. 8100 in3

C. 3240 in3

D. 4050 in3

Find the volume of the prism.

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Examples:

54

8

perimeter of base = 2(5) + 2(4) = 18

B = 5 x 4 = 20

L. A.= 18 x 8 = 144 sq. units

S.A. = 144 + 2(20) = 184 sq. units

V = 20 x 8 = 160 cubic units

h = 8

6

8

5

4

4

perimeter of base = 6 + 5 + 8 = 19

L. A. = 19 x 4 = 76 sq. units

B = ½ (6)(4) = 12

S. A. = 76 + 2(12) = 100 sq. units

V = 12 x 4 = 48 cubic units

h = 4

Prisms A and B have the same width and height, but different lengths. If the volume of Prism B is 128 cubic inches greater than the volume of Prism A,what is the length of each prism?

A 12 B 8

C 4 D 3.5

Prism APrism B

Read the Test ItemYou know the volume of each solid and that the difference between their volumes is 128 cubic inches.Solve the Test ItemVolume of Prism B –

Volume of Prism A = 128 Write an equation.4x ● 9 – 4x ● 5 = 128 Use V = Bh.

16x = 128 Simplify.x = 8 Divide each side by 16.

Answer: The length of each prism is 8 inches. The correct answer is B.

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Examples:

54

8

perimeter of base = 2(5) + 2(4) = 18

B = 5 x 4 = 20

L. A.= 18 x 8 = 144 sq. units

S.A. = 144 + 2(20) = 184 sq. units

V = 20 x 8 = 160 cubic units

h = 8

6

8

5

4

4

perimeter of base = 6 + 5 + 8 = 19

L. A. = 19 x 4 = 76 sq. units

B = ½ (6)(4) = 12

S. A. = 76 + 2(12) = 100 sq. units

V = 12 x 4 = 48 cubic units

h = 4

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Cylinders

2r

2r h

Surface Area (SA) = 2B + LA = 2πr ( r + h )

Cylinders are right prisms with circular bases.Therefore, the formulas for prisms can be used for cylinders.

Volume (V) = Bh =

The base area is the area of the circle:

The lateral area is the area of the rectangle: 2πrh

h

2πr

h

Formulas: S.A. = 2πr ( r + h )

V = 2r h

Lateral Area and Surface Area of a Cylinder

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

L = 2rh Lateral area of a cylinder

= 2(14)(18) Replace r with 14 and h with 18.

≈ 1583.4 Use a calculator.

Answer: The lateral area is about 1583.4 square feet and the surface area is about 2814.9 square feet.

S = 2rh + 2r2 Surface area of a cylinder

≈ 1583.4 + 2(14)2 Replace 2rh with 1583.4

and r with 14.

≈ 2814.9 Use a calculator.

A. lateral area ≈ 1508 ft2 andsurface area ≈ 2412.7 ft2

B. lateral area ≈ 1508 ft2 andsurface area ≈ 1206.4 ft2

C. lateral area ≈ 754 ft2 andsurface area ≈ 2412.7 ft2

D. lateral area ≈ 754 ft2 andsurface area ≈ 1206.4.7 ft2

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

Find Missing Dimensions

MANUFACTURING A soup can is covered with the label shown. What is the radius of the soup can?

L = 2rh Lateral area of a cylinder

125.6 = 2r(8) Replace L with 15.7 ● 8 and h with 8.

125.6 = 16r Simplify.

2.5 ≈ r Divide each side by 16.Answer: The radius of the soup can is about

2.5 inches.

A. 12 inches

B. 16 inches

C. 18 inches

D. 24 inches

Find the diameter of a base of a cylinder if the surface area is 480 square inches and the height is 8 inches.

Volume of a Cylinder

Find the volume of the cylinder to the nearest tenth.

Answer: The volume is approximately 18.3 cm3.

Volume of a cylinder

≈ 18.3 Use a calculator.

= (1.8)2(1.8) r = 1.8 and h = 1.8

A. 62.8 cm3

B. 628.3 cm3

C. 125.7 cm3

D. 1005.3 cm3

Find the volume of the cylinder to the nearest tenth.

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ExampleFor the cylinder shown, find the lateral area , surface area and volume.

L.A.= 2πr•h

L.A.= 2π(3)•(4)

L.A.= 24π sq. cm.

4 cm

3 cmS.A.= 2•πr2 + 2πr•h

S.A.= 2•π(3)2 + 2π(3)•(4)

S.A.= 18π +24π

S.A.= 42π sq. cm.V = πr2•h

V = π(3)2•(4)

V = 36π