Lesson 7.1.4

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Lesson 7.1.4Lesson 7.1.4

Surface Areas & Edge

Lengths of Complex Shapes

Surface Areas & Edge

Lengths of Complex Shapes

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Lesson

7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface Areas & Edge Lengths of Complex Shapes

California Standards:Measurement and Geometry 2.1Use 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 2.2Estimate 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 1.3Determine when and how to break a problem into simpler parts.

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

Measurement and Geometry 2.3Compute 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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Lesson


Prisms and cylinders can be stuck together to make complex shapes.

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.

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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Finding the Total Edge Length

Lesson


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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Example 1

Solution follows…

Lesson


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 in8 in

5 in

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Lesson


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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Example 2

Solution follows…

Lesson


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 20 cm

10 cm10 cm

5 cm

Solution continues…

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Example 2

Lesson



Solution (continued)

15 cm

20 cm 20 cm

10 cm10 cm

5 cm

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.


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Example 2

Lesson




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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Guided Practice

Solution follows…

Lesson


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

12 in

24 in

Cube = 144 in, Rectangular prism = 192 in

144 + 192 = 3366 ×12 = 72336 – 72 = 264 in

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Break Complex Figures Up to Work Out Surface Area

Lesson


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.

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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Example 3

Solution follows…

Lesson


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 cm1 cm

1 cm

8 cm

2 cm16 cm


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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


What is the surface area of this shape?1 cm

1 cm1 cm

8 cm

2 cm16 cm

The amount covered up on the big prism must be the same as the amount covered up on the small prism.



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Example 3


Lesson



The surface area of the bottom face of the

small prism is 1 cm2.

The surface area of the small prism is 6 cm2.

1 cm1 cm

1 cm

8 cm

2 cm16 cm

The surface area of the big prism is 352 cm2.


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Example 3

Lesson



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 cm2.

1 cm1 cm

1 cm

8 cm

2 cm16 cmSolution (continued)

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Guided Practice

Solution follows…

Lesson


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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Guided Practice

Solution follows…

Lesson


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 in20 in

5 in

5 in

4 in

14 + 18 = 3232 – 2 = 30 yd2

150 + 276.32 = 426.32426.32 – 25.12 = 401.2 in2

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Independent Practice

Solution follows…

Lesson


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 ft5 ft

3 ft

5 ft

40 ft and 44 ft

76 ft

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Independent Practice

Solution follows…

Lesson


Work out the surface areas of the shapes shown in Exercises 3–6. Use = 3.14.

3. 4.

5. 6.

24 in

12 in

12 in

12 in12 in

10 yd

20 yd

15 yd40 yd

10 yd

18 in 3 in

3 in2 in

18 in5 in

8 in 10 ft

10 ft

20 ft

7 ft10 ft

7 ft

Total height of shape = 24.9 ft

2016 in2

3699 yd2

2042 in2

917 ft2

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Round UpRound Up

Lesson


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.

Lesson 7.1.4

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