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**Lengths of Complex Shapes**

Surface Areas & Edge Lengths of Complex Shapes Lesson 7.1.4

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**7.1.4 California Standards: What it means for you: Key words:**

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. 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. What it means for you: You’ll work out the surface area and edge lengths of complex figures. Key words: net surface area edge prism

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

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**7.1.4 Finding the Total Edge Length**

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

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**7.1.4 Find the total edge length of the rectangular prism shown.**

Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Example 1 Find the total edge length of the rectangular prism shown. 6 in 8 in 5 in Solution There are four edges around the top face: = 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. So total edge length = = 76 in. Solution follows…

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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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Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Example 2 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? 15 cm 20 cm 10 cm 5 cm 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 Solution continues… Solution follows…

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Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Example 2 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? 15 cm 20 cm 10 cm 5 cm Solution (continued) But, two of the 10 cm edges on the top tier aren’t 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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Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Example 2 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) = – (2 × 10 × 2) = 280 cm So 280 cm of ribbon is needed.

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Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Guided Practice 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. Cube = 144 in, Rectangular prism = 192 in 24 in 12 in = 336 6 ×12 = 72 336 – 72 = 264 in Solution follows…

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**7.1.4 Break Complex Figures Up to Work Out Surface Area**

Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Break Complex Figures Up to Work Out Surface Area You work out the surface areas of complex shapes by breaking them into simple shapes and finding the surface area of each part. The place where two shapes are stuck together doesn’t form part of the complex shape’s surface — so you need to subtract the area of it from the areas of both simple shapes.

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**7.1.4 What is the surface area of this shape? Solution**

Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Example 3 1 cm 8 cm 2 cm 16 cm 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. Solution continues… Solution follows…

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**7.1.4 What is the surface area of this shape? Solution (continued)**

Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Example 3 1 cm 8 cm 2 cm 16 cm What is the surface area of this shape? Solution (continued) The amount covered up on the big prism must be the same as the amount covered up on the small prism. 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. Solution continues…

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**7.1.4 What is the surface area of this shape? Solution (continued)**

Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Example 3 1 cm 8 cm 2 cm 16 cm What is the surface area of this shape? Solution (continued) The surface area of the big prism is 352 cm2. The surface area of the small prism is 6 cm2. The surface area of the bottom face of the small prism is 1 cm2. Solution continues…

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**7.1.4 What is the surface area of this shape? Solution (continued)**

Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Example 3 1 cm 8 cm 2 cm 16 cm What is the surface area of this shape? Solution (continued) 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: – (2 × 1) = 356 cm2.

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

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Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Guided Practice In Exercises 6–7, work out the surface area of each shape. Use p = 3.14. 1 yd 3 yd 4 yd 5 in 20 in 4 in = – = in2 = 32 32 – 2 = 30 yd2 Solution follows…

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Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Independent Practice 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 40 ft and 44 ft 76 ft Solution follows…

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Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Independent Practice Work out the surface areas of the shapes shown in Exercises 3–6. Use p = 3.14. 24 in 12 in 10 yd 20 yd 15 yd 40 yd 2016 in2 3699 yd2 18 in 3 in 2 in 5 in 8 in 10 ft 20 ft 7 ft 917 ft2 Total height of shape = 24.9 ft 2042 in2 Solution follows…

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Lesson 7.1.4 Surface Areas & Edge Lengths of Complex Shapes Round Up So to find the total edge length of a complex shape, first break the shape up into simple shapes. 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. 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.

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