Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge...
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Transcript of Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge...
![Page 1: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/1.jpg)
Assignment
• P. 806-9: 2-20 even, 21, 24, 25, 28, 30
• P. 814-7: 2, 3-21 odd, 22-25, 30
• Challenge Problems: 3-5, 8, 9
![Page 2: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/2.jpg)
In Glorious 3-D!
Most of the figures you have worked with so far have been confined to a plane—two-dimensional. Solid figures in the “real world” have 3 dimensions: length, width, and height.
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Polyhedron
A solid formed by polygons that enclose a single region of space is called a polyhedron.
Separate your Geosolids into 2 groups: Polyhedra and others.
![Page 4: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/4.jpg)
Parts of Polyhedrons
• Polygonal region = face• Intersection of 2 faces = edge• Intersection of 3+ edges = vertex
face vertexedge
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Warm-Up
Separate your Geosolid polyhedra into two groups where each of the groups have similar characteristics.
What are the names of these groups?
1. Prisms2. Pyramids
Polyhedra:
![Page 6: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/6.jpg)
12.2 & 12.3: Surface Area of Prisms, Cylinders, Pyramids, and Cones
Objectives:
1. To find and use formulas for the lateral and total surface area of prisms, cylinders, pyramids, and cones
![Page 7: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/7.jpg)
Prism
A polyhedron is a prism iff it has two congruent parallel bases and its lateral faces are parallelograms.
![Page 8: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/8.jpg)
Classification of Prisms
Prisms are classified by their bases.
![Page 9: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/9.jpg)
Right & Oblique Prisms
Prisms can be right or oblique. What differentiates the two?
![Page 10: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/10.jpg)
Right & Oblique Prisms
In a right prism, the lateral edges are perpendicular to the base.
![Page 11: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/11.jpg)
Pyramid
A polyhedron is a pyramid iff it has one base and its lateral faces are triangles with a common vertex.
![Page 12: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/12.jpg)
Classification of Pyramids
Pyramids are also classified by their bases.
![Page 13: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/13.jpg)
Pyramid
A regular pyramid is one whose base is a regular polygon.
![Page 14: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/14.jpg)
Pyramid
A regular pyramid is one whose base is a regular polygon.
• The slant height is the height of one of the congruent lateral faces.
![Page 15: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/15.jpg)
Solids of Revolution
The three-dimensional figure formed by spinning a two dimensional figure around an axis is called a solid of revolution.
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Cylinder
A cylinder is a 3-D figure with two congruent and parallel circular bases.
• Radius = radius of base
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Cone
A cone is a 3-D figure with one circular base and a vertex not on the same plane as the base.
• Altitude = perpendicular segment connecting vertex to the plane containing the base (length = height)
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Cone
A cone is a 3-D figure with one circular base and a vertex not on the same plane as the base.
• Slant height = segment connecting vertex to the circular edge of the base
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Right vs. Oblique
What is the difference between a right and an oblique cone?
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Right vs. Oblique
In a right cone, the segment connecting the vertex to the center of the base is perpendicular to the base.
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Nets
Imagine cutting a 3-D solid along its edges and laying flat all of its surfaces. This 2-D figure is a net for that 3-D solid.
An unfolded pizza box is a net!
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Nets
Imagine cutting a 3-D solid along its edges and laying flat all of its surfaces. This 2-D figure is a net for that 3-D solid.
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Activity: Red, Rubbery Nets
Match one of the red, rubbery nets with its corresponding 3-D solid. Which of the shapes has no net?
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Activity: Red, Rubbery Nets
Match one of the red, rubbery nets with its corresponding 3-D solid. Which of the shapes has no net?
A sphere doesn’t have a true net; it can only be approximated.
![Page 25: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/25.jpg)
Exercise 1
There are generally two types of measurements associated with 3-D solids: surface area and volume. Which of these can be easily found using a shape’s net?
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Surface Area
The surface area of a 3-D figure is the sum of the areas of all the faces or surfaces that enclose the solid.
• Asking how much surface area a figure has is like asking how much wrapping paper it takes to cover it.
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Lateral Surface Area
The lateral surface area of a 3-D figure is the sum of the areas of all the lateral faces of the solid.
• Think of the lateral surface area as the size of a label that you could put on the figure.
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Exercise 2
What solid corresponds to the net below?
How could you find the lateral and total surface area?
![Page 29: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/29.jpg)
Exercise 3
Draw a net for the rectangular prism below.
To find the lateral surface area, you could:
• Add up the areas of the lateral rectangles
A B C D
![Page 30: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/30.jpg)
Exercise 3
Draw a net for the rectangular prism below.
To find the lateral surface area, you could:
• Find the area of the lateral surface as one, big rectangle
Hei
gh
t o
f P
rism
Perimeter of the Base
![Page 31: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/31.jpg)
Exercise 3
Draw a net for the rectangular prism below.
To find the total surface area, you could:
• Find the lateral surface area then add the two bases
Hei
gh
t o
f P
rism
Perimeter of the Base
![Page 32: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/32.jpg)
Surface Area of a Prism
Lateral Surface Area of a Prism:
• P = perimeter of the base• h = height of the prism
Total Surface Area of a Prism:
• B = area of the base
S Ph 2S Ph B
![Page 33: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/33.jpg)
Exercise 4
Find the lateral and total surface area.
![Page 34: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/34.jpg)
Exercise 5
Draw a net for the cylinder.
Notice that the lateral surface of a cylinder is also a rectangle. Its height is the height of the cylinder, and the base is the circumference of the base.
![Page 35: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/35.jpg)
Exercise 6
Write formulas for the lateral and total surface area of a cylinder.
![Page 36: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/36.jpg)
Surface Area of a Cylinder
Lateral Surface Area of a Cylinder:
• C = circumference of base• r = radius of base• h = height of the cylinder
Total Surface Area of a Cylinder:
S Ch2S Ph B
2S rh
S Ph22 2S rh r
2S r h r
![Page 37: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/37.jpg)
Exercise 7
The net can be folded to form a cylinder.
What is the approximate lateral and total surface area of the cylinder?
![Page 38: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/38.jpg)
Height vs. Slant Height
By convention, h represents height and l represents slant height.
![Page 39: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/39.jpg)
Height vs. Slant Height
By convention, h represents height and l represents slant height.
![Page 40: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/40.jpg)
Exercise 8
Draw a net for the square pyramid below.
To find the lateral surface area:
• Find the area of one triangle, then multiply by 4
12 40 25A
124 40 25S
12 4 40 25S
![Page 41: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/41.jpg)
Exercise 8
Draw a net for the square pyramid below.
To find the lateral surface area:
• Find the area of one triangle, then multiply by 4
12 40 25A
124 40 25S
12 4 40 25S
12 4S s l
12S Pl
![Page 42: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/42.jpg)
Exercise 8
Draw a net for the square pyramid below.
To find the total surface area:
• Just add the area of the base to the lateral area
12 40 25A
124 40 25S
12 4 40 25S
12 4S s l
12S Pl12S Pl B
![Page 43: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/43.jpg)
Surface Area of a Pyramid
Lateral Surface Area of a Pyramid:
• P = perimeter of the base• l = slant height of the
pyramid
Total Surface Area of a Prism:
• B = area of the base
12S Pl 1
2S Pl B
![Page 44: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/44.jpg)
Exercise 9
Find the lateral and total surface area.
![Page 45: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/45.jpg)
Exercise 10
You may have realized that the formula for the lateral area for a prism and a cylinder are basically the same. The same is true for the formulas for a pyramid and a cone. Derive a formula for the lateral area of a cone.
12S Pl
Lateral area of a Pyramid: Lateral area of a Cone:12S Cl
12 2S r l
S rl
![Page 46: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/46.jpg)
Surface Area of a Cone
Lateral Surface Area of a Cone:
• r = radius of the base• l = slant height of the
cone
Total Surface Area of a Cone:
S rl 2S rl r
S r l r
![Page 47: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/47.jpg)
Exercise 11
A traffic cone can be approximated by a right cone with radius 5.7 inches and height 18 inches. To the nearest tenth of a square inch, find the approximate lateral area of the traffic cone.
![Page 48: Assignment P. 806-9: 2-20 even, 21, 24, 25, 28, 30 P. 814-7: 2, 3-21 odd, 22-25, 30 Challenge Problems: 3-5, 8, 9.](https://reader035.fdocuments.in/reader035/viewer/2022062222/5697bf8a1a28abf838c8a696/html5/thumbnails/48.jpg)
Tons of Formulas?
Really there’s just two formulas, one for prisms/cylinders and one for pyramids/cones.
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Assignment
• P. 806-9: 2-20 even, 21, 24, 25, 28, 30
• P. 814-7: 2, 3-21 odd, 22-25, 30
• Challenge Problems: 3-5, 8, 9