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

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

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

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

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

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

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

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

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. 33 10 6 6 10 33 = Surface area = (6 x 10) + (6 x 10) + (6 x 10) + ½(6)(33) + ½ (6)(33) = 60 + 60 + 60 + 93 + 93 = 180 + 183 Lesson 10-2: 3-D Views of Solid Figures

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

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

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.

Find the lateral area of the regular octagonal prism. A. 162 cm2 B. 216 cm2 C. 324 cm2 D. 432 cm2

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

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.

Find the surface area of the triangular prism. A. 320 units2 B. 512 units2 C. 368 units2 D. 416 units2

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.

Find the volume of the prism. A. 6480 in3 B. 8100 in3 C. 3240 in3 D. 4050 in3

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 = 6 + 5 + 8 = 19 L. A.= 18 x 8 = 144 sq. units L. A. = 19 x 4 = 76 sq. units S.A. = 144 + 2(20) = 184 sq. units S. A. = 76 + 2(12) = 100 sq. units V = 20 x 8 = 160 cubic units V = 12 x 4 = 48 cubic units

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

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.

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 = 6 + 5 + 8 = 19 L. A.= 18 x 8 = 144 sq. units L. A. = 19 x 4 = 76 sq. units S.A. = 144 + 2(20) = 184 sq. units S. A. = 76 + 2(12) = 100 sq. units V = 20 x 8 = 160 cubic units V = 12 x 4 = 48 cubic units

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

L = 2rh 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 = 2rh Lateral area of a cylinder = 2(14)(18) Replace r with 14 and h with 18. ≈ 1583.4 Use a calculator.

S = 2rh + 2r2 Surface area of a cylinder ≈ 1583.4 + 2(14)2 Replace 2rh with 1583.4 and r with 14. ≈ 2814.9 Use a calculator. Answer: The lateral area is about 1583.4 square feet and the surface area is about 2814.9 square feet.

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 ≈ 2412.7 ft2 B. lateral area ≈ 1508 ft2 and surface area ≈ 1206.4 ft2 C. lateral area ≈ 754 ft2 and surface area ≈ 2412.7 ft2 D. lateral area ≈ 754 ft2 and surface area ≈ 1206.4.7 ft2

L = 2rh 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 = 2rh Lateral area of a cylinder 125.6 = 2r(8) Replace L with 15.7 ● 8 and h with 8. 125.6 = 16r Simplify. 2.5 ≈ r Divide each side by 16. Answer: The radius of the soup can is about 2.5 inches.

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

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.

Find the volume of the cylinder to the nearest tenth. A. 62.8 cm3 B. 628.3 cm3 C. 125.7 cm3 D. 1005.3 cm3

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