Assignment
description
Transcript of Assignment
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
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.
PolyhedronA solid formed by polygons that enclose a
single region of space is called a polyhedron.
Separate your Geosolids into 2 groups: Polyhedra and others.
Parts of Polyhedrons• Polygonal region = face• Intersection of 2 faces = edge• Intersection of 3+ edges = vertex
face vertexedge
Warm-UpSeparate 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:
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
PrismA polyhedron is a prism
iff it has two congruent parallel bases and its lateral faces are parallelograms.
Classification of PrismsPrisms are classified by their bases.
Right & Oblique PrismsPrisms can be right or oblique. What
differentiates the two?
Right & Oblique PrismsIn a right prism, the lateral edges are
perpendicular to the base.
PyramidA polyhedron is a pyramid iff it has one base
and its lateral faces are triangles with a common vertex.
Classification of PyramidsPyramids are also classified by their bases.
PyramidA regular pyramid is one whose base is a
regular polygon.
PyramidA regular pyramid is one whose base is a
regular polygon.
• The slant height is the height of one of the congruent lateral faces.
Solids of RevolutionThe three-dimensional figure formed by
spinning a two dimensional figure around an axis is called a solid of revolution.
CylinderA cylinder is a 3-D
figure with two congruent and parallel circular bases.
• Radius = radius of base
ConeA 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)
ConeA 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
Right vs. ObliqueWhat is the difference between a right and
an oblique cone?
Right vs. ObliqueIn a right cone, the segment connecting the vertex to
the center of the base is perpendicular to the base.
NetsImagine 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!
NetsImagine 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.
Activity: Red, Rubbery Nets
Match one of the red, rubbery nets with its corresponding 3-D solid. Which of the shapes has no net?
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.
Exercise 1There 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?
Surface AreaThe 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.
Lateral Surface AreaThe 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.
Exercise 2What solid corresponds to the net below?
How could you find the lateral and total surface area?
Exercise 3Draw 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
Exercise 3Draw 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
ght o
f Pris
m
Perimeter of the Base
Exercise 3Draw 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
ght o
f Pris
m
Perimeter of the Base
Surface Area of a PrismLateral 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
Exercise 4Find the lateral and total surface area.
Exercise 5Draw 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.
Exercise 6Write formulas for the lateral and total surface area
of a cylinder.
Surface Area of a CylinderLateral 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
Exercise 7The net can be
folded to form a cylinder.
What is the approximate lateral and total surface area of the cylinder?
Height vs. Slant HeightBy convention, h represents height and l
represents slant height.
Height vs. Slant HeightBy convention, h represents height and l
represents slant height.
Exercise 8Draw 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
Exercise 8Draw 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
Exercise 8Draw 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
Surface Area of a PyramidLateral 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
Exercise 9Find the lateral and total surface area.
Exercise 10You 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
Surface Area of a ConeLateral 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
Exercise 11A 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.
Tons of Formulas?Really there’s just two formulas, one for
prisms/cylinders and one for pyramids/cones.
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