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**Lateral Area, Surface Area, and Volume**

Honors Geometry Unit 8 Prisms and Cylinders Lesson 2 Lateral Area, Surface Area, and Volume

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

Different Views Perspective view of a cone Different angle views of a cone the side (or from any side view) the top the bottom Lesson 10-2: 3-D Views of Solid Figures

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**Example: Different Views**

Front Left Right Back Top * Note: The dark lines indicated a break in the surface. Lesson 10-2: 3-D Views of Solid Figures

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

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

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

Sketches - continued Step 2: Draw segments 3 units down from each vertex (show hidden sides with dotted lines). Lesson 10-2: 3-D Views of Solid Figures

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

Sketches - continued 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. Lesson 10-2: 3-D Views of Solid Figures

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

Nets and Surface Area Imagine cutting a cardboard box along its edges and laying it out flat. The resulting figure is called a net. top back end front bottom = A net is very helpful in finding the surface area of a solid figure. 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.**

First, imagine the figure represented as a net. Find the area of each face. Find the sum of all the individual areas. 33 10 6 6 10 33 = Surface area = (6 x 10) + (6 x 10) + (6 x 10) + ½(6)(33) + ½ (6)(33) = 3 + 93 = 3 Lesson 10-2: 3-D Views of Solid Figures

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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 Triangular Prism h

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**Find the lateral area of the regular hexagonal 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. Lateral area of a prism P = 30, h = 12 Multiply. Answer: The lateral area is 360 square centimeters.

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**Find the lateral area of the regular octagonal prism.**

A. 162 cm2 B. 216 cm2 C. 324 cm2 D. 432 cm2

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**Find the surface area of the rectangular prism.**

Surface Area of a Prism Find the surface area of the rectangular prism.

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**Answer: The surface area is 360 square centimeters.**

Surface Area of a Prism Surface area of a prism L = Ph Substitution Simplify. Answer: The surface area is 360 square centimeters.

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**Find the surface area of the triangular prism.**

A. 320 units2 B. 512 units2 C. 368 units2 D. 416 units2

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**Find the volume of the prism.**

Volume of a Prism Find the volume of the prism. V Bh Volume of a prism 1500 Simplify. Answer: The volume of the prism is 1500 cubic centimeters.

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**Find the volume of the prism.**

A in3 B in3 C in3 D in3

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**Examples: 5 4 8 h = 8 h = 4 B = 5 x 4 = 20 B = ½ (6)(4) = 12**

perimeter of base = 2(5) + 2(4) = 18 perimeter of base = = 19 L. A.= 18 x 8 = 144 sq. units L. A. = 19 x 4 = 76 sq. units S.A. = (20) = 184 sq. units S. A. = (12) = 100 sq. units V = 20 x 8 = 160 cubic units V = 12 x 4 = 48 cubic units

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**Prisms A and B have the same width and height, but different lengths**

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? Prism B Prism A A 12 B 8 C 4 D 3.5

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Read the Test Item You know the volume of each solid and that the difference between their volumes is 128 cubic inches. Solve the Test Item Volume 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: 5 4 8 h = 8 h = 4 B = 5 x 4 = 20 B = ½ (6)(4) = 12**

perimeter of base = 2(5) + 2(4) = 18 perimeter of base = = 19 L. A.= 18 x 8 = 144 sq. units L. A. = 19 x 4 = 76 sq. units S.A. = (20) = 184 sq. units S. A. = (12) = 100 sq. units V = 20 x 8 = 160 cubic units V = 12 x 4 = 48 cubic units

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**Cylinders Formulas: S.A. = 2πr ( r + h ) V =**

Cylinders are right prisms with circular bases. Therefore, the formulas for prisms can be used for cylinders. Surface Area (SA) = 2B + LA = 2πr ( r + h ) The base area is the area of the circle: The lateral area is the area of the rectangle: 2πrh Volume (V) = Bh = h 2πr h

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**L = 2rh Lateral area of a cylinder **

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. ≈ Use a calculator.

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

≈ (14)2 Replace 2rh with and r with 14. ≈ Use a calculator. Answer: The lateral area is about square feet and the surface area is about square feet.

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**Find the lateral area and the surface area of the cylinder**

Find the lateral area and the surface area of the cylinder. Round to the nearest tenth. A. lateral area ≈ 1508 ft2 and surface area ≈ ft2 B. lateral area ≈ 1508 ft2 and surface area ≈ ft2 C. lateral area ≈ 754 ft2 and surface area ≈ ft2 D. lateral area ≈ 754 ft2 and surface area ≈ ft2

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**L = 2rh Lateral area of a cylinder **

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 ● 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.

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Find the diameter of a base of a cylinder if the surface area is 480 square inches and the height is 8 inches. A. 12 inches B. 16 inches C. 18 inches D. 24 inches

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**Find the volume of the cylinder to the nearest tenth.**

Volume of a Cylinder Find the volume of the cylinder to the nearest tenth. Volume of a cylinder = (1.8)2(1.8) r = 1.8 and h = 1.8 ≈ 18.3 Use a calculator. Answer: The volume is approximately 18.3 cm3.

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**Find the volume of the cylinder to the nearest tenth.**

A cm3 B cm3 C cm3 D cm3

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Example For the cylinder shown, find the lateral area , surface area and volume. 3 cm S.A.= 2•πr2 + 2πr•h L.A.= 2πr•h 4 cm S.A.= 2•π(3)2 + 2π(3)•(4) L.A.= 2π(3)•(4) S.A.= 18π +24π L.A.= 24π sq. cm. S.A.= 42π sq. cm. V = πr2•h V = π(3)2•(4) V = 36π

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