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1 Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Surface Areas & Edge Lengths of Complex Shapes

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2 Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes California Standards: Measurement and Geometry 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. Measurement and Geometry 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. Mathematical Reasoning 1.3 Determine when and how to break a problem into simpler parts. What it means for you: Youll work out the surface area and edge lengths of complex figures. Key words: net surface area edge prism Measurement and Geometry 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.

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3 Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Prisms and cylinders can be stuck together to make complex shapes. You can use lots of the skills youve already learned to find the total edge length and surface area of a complex shape but there are some important things to watch for. For example, a house might be made up of a rectangular prism with a triangular prism on top for the roof.

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4 Finding the Total Edge Length Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes The tricky thing about finding the total edge length of a solid, is making sure that you include each edge length only once. An edge on a solid shape is a line where two faces meet. An edge

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5 Example 1 Solution follows… Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Find the total edge length of the rectangular prism shown. Solution So total edge length = 28 + 28 + 20 = 76 in. There are four edges around the top face: 6 + 6 + 8 + 8 = 28 in. The bottom is identical to the top, so this also has an edge length of 28 in. There are four vertical edges joining the top and bottom: 4 × 5 = 20 in. 6 in 8 in 5 in

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6 Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes When complex shapes are formed from simple shapes, some of the edges of the simple shapes might disappear. These need to be subtracted. When two rectangular prisms are joined… …some edges disappear.

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7 Example 2 Solution follows… Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes A wedding cake has two tiers. The back and front views are shown below. The cake is to have ribbon laid around its edges. What is the total length of ribbon needed? Solution Total edge length of the top tier: (10 × 8) + (5 × 4) = 100 cm Total edge length of the bottom tier: (20 × 8) + (15 × 4) = 220 cm 15 cm 20 cm 10 cm 5 cm Solution continues…

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8 Example 2 Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes A wedding cake has two tiers. The back and front views are shown below. The cake is to have ribbon laid around its edges. What is the total length of ribbon needed? Solution (continued) 15 cm 20 cm 10 cm 5 cm But, two of the 10 cm edges on the top tier arent edges on the finished cake. You have to subtract these shared edge lengths from both the top and the bottom. Solution continues…

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9 Example 2 Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes A wedding cake has two tiers. The back and front views are shown below. The cake is to have ribbon laid around its edges. What is the total length of ribbon needed? Solution (continued) Total edge length = top tier edge lengths + bottom tier edge lengths – (2 × shared edge lengths) = 100 + 220 – (2 × 10 × 2) = 280 cm So 280 cm of ribbon is needed.

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10 Guided Practice Solution follows… Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes A display stand is formed from a cube and a rectangular prism. 1. Find the total edge lengths of the cube and rectangular prism before they were joined. 2. Find the total edge length of the display stand. 24 in 12 in 24 in Cube = 144 in, Rectangular prism = 192 in 144 + 192 = 336 6 ×12 = 72 336 – 72 = 264 in

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11 Break Complex Figures Up to Work Out Surface Area Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes The place where two shapes are stuck together doesnt form part of the complex shapes surface so you need to subtract the area of it from the areas of both simple shapes. You work out the surface areas of complex shapes by breaking them into simple shapes and finding the surface area of each part.

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12 Example 3 Solution follows… Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes What is the surface area of this shape? Solution The complex shape is like two rectangular prisms stuck together. You can work out the surface area of each individually, and then add them together. But the bottom of the small prism is covered up, as well as some of the top of the large prism. So you lose some surface area. 1 cm 8 cm 2 cm 16 cm Solution continues…

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13 So you have to subtract the area of the bottom face of the small prism twice once to take away the face on the small prism, and once to take away the same shape on the big prism. Example 3 Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes What is the surface area of this shape? 1 cm 8 cm 2 cm 16 cm The amount covered up on the big prism must be the same as the amount covered up on the small prism. Solution continues… Solution (continued)

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14 Example 3 Solution continues… Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes What is the surface area of this shape? The surface area of the bottom face of the small prism is 1 cm 2. The surface area of the small prism is 6 cm 2. 1 cm 8 cm 2 cm 16 cm The surface area of the big prism is 352 cm 2. Solution (continued)

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15 Example 3 Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes What is the surface area of this shape? Total surface area = surface area of big prism + surface area of small prism – (2 × bottom face of small prism). So the surface area of the shape is: 352 + 6 – (2 × 1) = 356 cm 2. 1 cm 8 cm 2 cm 16 cm Solution (continued)

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16 Guided Practice Solution follows… Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes In Exercises 3–5, suggest how the complex figure could be split up into simple figures. 3. 4. 5. Two, three, or four rectangular prisms Four cylinders and a rectangular prism Three cylinders

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17 Guided Practice Solution follows… Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes In Exercises 6–7, work out the surface area of each shape. Use = 3.14. 6. 7. 1 yd 3 yd 1 yd 4 yd 1 yd 5 in 20 in 5 in 4 in 14 + 18 = 32 32 – 2 = 30 yd 2 150 + 276.32 = 426.32 426.32 – 25.12 = 401.2 in 2

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18 Independent Practice Solution follows… Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes A kitchen work center is made from two rectangular prisms. It is to have a trim around the edge. 1. Find the total edge length of each of the prisms separately. 2. Find the length of trim needed for the work center. 3 ft 2 ft 5 ft 3 ft 5 ft 40 ft and 44 ft 76 ft

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19 Independent Practice Solution follows… Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Work out the surface areas of the shapes shown in Exercises 3–6. Use = 3.14. 3. 4. 5. 6. 24 in 12 in 10 yd 20 yd 15 yd 40 yd 10 yd 18 in 3 in 2 in 18 in 5 in 8 in 10 ft 20 ft 7 ft 10 ft 7 ft Total height of shape = 24.9 ft 2016 in 2 3699 yd 2 2042 in 2 917 ft 2

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20 Round Up Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes So to find the total edge length of a complex shape, first break the shape up into simple shapes. The surface area of complex shapes is found the same way. Remember the edge lengths and surface areas that disappear need to be subtracted twice once from each shape. Then you can find the edge length of each piece separately. But then you have to think about which edges disappear when the complex shape is made.

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