lesson 7.1.4
DESCRIPTION
Surface Areas & Edge Lengths of Complex Shapes. Lesson 7.1.4. Lesson 7.1.4. Surface Areas & Edge Lengths of Complex Shapes. California Standards: Measurement and Geometry 2.1 - PowerPoint PPT PresentationTRANSCRIPT
1
Lesson 7.1.4Lesson 7.1.4
Surface Areas & Edge
Lengths of Complex Shapes
Surface Areas & Edge
Lengths of Complex Shapes
2
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.
3
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface Areas & Edge Lengths of Complex Shapes
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.
4
Finding the Total Edge Length
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface 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
5
Example 1
Solution follows…
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface 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 in8 in
5 in
6
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface 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.
7
Example 2
Solution follows…
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface 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 20 cm
10 cm10 cm
5 cm
Solution continues…
8
Example 2
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface 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 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.
Solution continues…
9
Example 2
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface 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.
10
Guided Practice
Solution follows…
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface 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
12 in
24 in
Cube = 144 in, Rectangular prism = 192 in
144 + 192 = 3366 ×12 = 72336 – 72 = 264 in
11
Break Complex Figures Up to Work Out Surface Area
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface Areas & Edge Lengths of Complex Shapes
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.
12
Example 3
Solution follows…
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface 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 cm1 cm
1 cm
8 cm
2 cm16 cm
Solution continues…
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
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface Areas & Edge Lengths of Complex Shapes
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.
Solution continues…
Solution (continued)
14
Example 3
Solution continues…
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface 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 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.
Solution (continued)
15
Example 3
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface 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 cm2.
1 cm1 cm
1 cm
8 cm
2 cm16 cmSolution (continued)
16
Guided Practice
Solution follows…
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface 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
17
Guided Practice
Solution follows…
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface 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 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
18
Independent Practice
Solution follows…
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface 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 ft5 ft
3 ft
5 ft
40 ft and 44 ft
76 ft
19
Independent Practice
Solution follows…
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface 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
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
20
Round UpRound Up
Lesson
7.1.4Surface Areas & Edge Lengths of Complex ShapesSurface 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.