Upload
marshall-carroll
View
212
Download
0
Tags:
Embed Size (px)
Citation preview
CONFIDENTIAL 1
GeometryGeometry
Review Solid Review Solid GeometryGeometry
CONFIDENTIAL 2
Warm UpWarm Up
Determine whether the two polygons are similar. If so, give the similarity ratio.
1) 2)
8
8
2 2
12
4
12
4
11.942.5
40.8
7
24
25
CONFIDENTIAL 3
Solid GeometrySolid Geometry
Three-dimensional figures, or solids, can be made up of flat or curved surfaces. Each flat surface is called a face. An edge is the segment that is the
intersection of two faces. A vertex is the point that is the intersection of three or more faces.
Face Edge
Vertex
CONFIDENTIAL 4
Three-Dimensional Figures Three-Dimensional Figures
TERM EXAMPLE
A Prism is formed by two parallel congruent polygonal faces called bases
connected by faces that are parallelograms.
Bases
A cylinder is formed by two parallel congruent circular bases and curved
surface that connects the bases. Bases
CONFIDENTIAL 5
TERM EXAMPLE
A pyramid is formed by a polygonal base and triangular faces that meet at
a common vertex.
Vertex
Base
A cone is formed by a circular base and a curved surface that connects
the base to a vertex.Base
Vertex
CONFIDENTIAL 6
A cube is a prism with six square faces. Other prisms and pyramids are named for the shape of their bases.
TriangularPrism
RectangularPrism
PentagonalPrism
HexagonalPrism Next Page:
CONFIDENTIAL 7
Triangularpyramid
Rectangularpyramid
Pentagonalpyramid
Hexagonalpyramid
CONFIDENTIAL 8
Classifying Three-Dimensional Figures
Classify the figure. Name the vertices, edges, and bases.
A.
A
B C
D
E
Rectangular pyramid
Rectangular pyramid
Vertices: A,B,C,D,E
Edges: AB, BC, CD, AD, AE,BE, CE, DE
Base: rectangle ABCD
Next Page:
CONFIDENTIAL 9
Identifying a Three-Dimensional Figure From a Net
Describe the three-dimensional figure that can be made from the given net.
A)
The net has two congruent triangular faces. The remaining faces are parallelograms, so the net forms a triangular prism.
CONFIDENTIAL 10
Describing Cross Sections of Three-Dimensional Figures
Describe the cross section.
A The cross section is a triangle.
CONFIDENTIAL 11
Food Application
A chef is slicing a cube-shaped watermelon for a buffet. How can the chef cut the watermelon to
make a slice of each shape?
A A square
Cut parallel to the bases.
CONFIDENTIAL 12
Now you try!
1) Classify each figure. Name the vertices, edges, and bases.
a)
B
Ab)
K
G
J
DC
EF
H
CONFIDENTIAL 13
Representations of Three-Dimensional Figures Representations of Three-Dimensional Figures
There are many ways to represent a three-dimensional object. An orthographic drawing shows six different views of an object: top, bottom, front, back, left side, and right side.
Top
Back
Bottom
Front
Left
Right
CONFIDENTIAL 14
Drawing Orthographic Views of an Drawing Orthographic Views of an ObjectObject
Draw all six orthographic views of the given object.Assume there are no hidden cubes.
Front:
Top:
Back:
Right:Left:
Bottom:
CONFIDENTIAL 15
Isometric drawing is a way to show three sides of a figure from a corner view. You can use isometric dot paper to make an isometric drawing. This paper has diagonal rows of dots that are equally spaced in a repeating triangular pattern.
CONFIDENTIAL 16
Drawing an isometric View of an Object
Draw an isometric view of the given object. assume there are no hidden cubes.
CONFIDENTIAL 17
In a perspective drawing, nonvertical parallel lines are drawn so that they meet at a point called a vanishing point. Vanishing point are located on a horizontal line called the horizon. A one-point perspective drawing contains one vanishing points. A two-point perspective drawing contains two vanishing points.
Vanishing point s
Vanishing point
one-point perspective
two-point perspective
CONFIDENTIAL 18
Drawing an Object in Perspective
3 A) Draw a cube in one-point perspective.
Draw a horizontal line to represent the horizon. Mark a vanishing point on the horizon. This is the front of the cube.
From each corner of the square, lightly draw dashed segments to the vanishing point.
Next page
CONFIDENTIAL 19
Lightly draw a smaller square with vertices on the dashed segments. This is the back of the cube.
Draw the edges of the cube, using dashed segments for hidden edges. Erase any segments that are not part of the cube.
Next page
CONFIDENTIAL 20
Relating Different Representations of an Object
4) Determine whether each drawing represents the given object. Assume there are no hidden cubes.
A B
Yes; the drawing is a one-point perspective view of the object.
No; the figure in the drawing is made up of four cubes, and the object is made up of only three cubes.
Next page
CONFIDENTIAL 21
Now you try!
1) Determine whether drawing represents the given object. Assume there are no hidden cubes.
Front
Top
BackRightLeft
Bottom
CONFIDENTIAL 22
Formulas in Three Dimensions
A polyhedron is formed by four or more polygons that intersect only at their edges. Prism and pyramids are
polyhedrons, but cylinders and cones are not.
polyhedrons Not polyhedron
CONFIDENTIAL 23
A polyhedron is a solid object whose surface is made up of a number of flat faces which themselves are bordered by straight lines. Each face is in fact a polygon, a closed shape in the flat 2-dimensional plane made up of points
joined by straight lines.
The familiar triangle and square are both polygons, but polygons can also have more irregular shapes like the one shown on the right.
A polygon is called regular if all of its sides are the same length, and all the angles between them are the same.
CONFIDENTIAL 24
A polyhedron is what you get when you move one dimension up. It is a closed, solid object whose surface is made up of a number of polygonal faces. We call the sides of these faces edges — two faces meet along each one of these edges. We call the corners of the faces vertices, so that any vertex lies on at least three
different faces. To illustrate this, here are two examples of well-known polyhedra.
The familiar cube on the left and the icosahedrons on the right. A polyhedron consists of polygonal faces, their sides are known as
edges, and the corners as vertices.
CONFIDENTIAL 25
A polyhedron consists of just one piece. It cannot, for example, be made up of two (or more) basically separate parts joined by
only an edge or a vertex. This means that neither of the following objects is a true polyhedron.
These objects are not polyhedra because they are made up of two separate parts meeting only in an edge (on the left) or a vertex (on the right).
CONFIDENTIAL 26
Euler’s Formula
For any polyhedron with V vertices, E edges, and F faces,
V - E + F = 2.
CONFIDENTIAL 27
Using Euler’s FormulaFind the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula.
A B
Using Euler’s Formula.Simplify
V= 4, E = 6, F = 44 - 6 + 4 = 2
2=2
V = 10, E = 15, F =710 - 15 + 7 = 2
2=2
CONFIDENTIAL 28
A diagonal of a three-dimensional figure connects
two vertices of two different faces. Diagonal d of a
rectangular prism is shown in the diagram. By the
PythagoreanTheoram, l + w = x and x + h = d.
Using substitution, l + w + h = d.
22 22
2
2
2 22
2
l
hd
wx
CONFIDENTIAL 29
Diagonal of a Right Rectangular Prism
The length of a diagonal d of a right rectangular prism with length l , width w, and height h is
d = 2 22
l + w + h .
CONFIDENTIAL 30
Space is the set of all points in three dimensions. Three coordinates are needed to locate a point in space. A three-dimensional coordinate system has 3 perpendicular axes: the x-axis, the y-axis, and the z-axis. An ordered triple (x , y ,z) is used to located a point. To located the point (3, 2, 4), start at (0, 0, 0). From there move 3 units forward, 2 units right, and then 4 units up
x
y
z8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
(3,2,4)
2
4
3
CONFIDENTIAL 31
Graphing Figures in Three Graphing Figures in Three DimensionsDimensions
Graph the figure.
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
(0, 0, 0)
(0, 0, 4)
(4, 0, 4)
(4, 0, 0)
x
z
y
(4, 4, 4)(0, 4, 4)
(0, 4, 0)
(4, 4, 0)
A) A cubed with edge length 4 units and one vertex at (0, 0, 0)
The cube has 8 vertices:(0, 0, 0), (0, 4, 0),(0, 0, 4), (4, 0, 0)(4, 4, 0), (4, 0, 4), (0, 4, 4),(4, 4, 4)
Next page
CONFIDENTIAL 32
z
x
y
M(x2, y2, z2) (x1, y1, z1)
You can find the distance between the two points (x1, y1, z1)and (x2, y2, z2) by drawing a rectangular prism with the given points as endpoints of a diagonal. Then use the formula for the length of the diagonal. You can also use a formula related to the Distance formula. (see Lesson 1-6.) The formula for the midpoint between (x1, y1, z1) and (x2, y2, z1) is related to the Midpoint formula. (see Lesson 1-6.)
CONFIDENTIAL 33
Distance and Midpoint Formulas in three Dimensions
The distance between the points (x1, y1, z1) and (x2, y2, z2) is d = (x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2 .
The midpoint of the segment with endpoints (x1, y1, z1) and(x2, y2, z2) is
M(x1 + x2)
2,
(y1+ y2)
2,
(z1 + z2)
2 .
CONFIDENTIAL 34
Finding Distances and Midpoints in Three Dimensions
Find the distance between the given points. Find the midpoint of the segment with the given endpoints.
Round to the nearest tenth, if necessary.
A) (0, 0, 0) and (3, 4, 12)
Distance: Midpoint:
d = (x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2
= (3 - 0)2 + (4 - 0)2 + (12 - 0)2
= 9 + 16 + 144 = 169 = 13 units
M(x1 + x2)
2,
(y1+ y2)
2,
(z1 + z2)
2
M0 + 3
2,
0 + 4
2,
0 + 12
2
M(1.5, 2, 6)
CONFIDENTIAL 35
Recreation Application
Two divers swam from a boat to the locations shown in the diagram. How far apart are the divers?
9 ft
Depth: 8 ft
Depth: 12 ft
18 ft
15 ft
6 ft
The location of the boat can be represented by the ordered triple (0, 0, 0), and the location of the divers can be represented by the ordered triples (18, 9, -8) and (-15, -6, -12).
d = (x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2
= (-15 - 18)2 + (-6 - 9)2 + (-12+ 8)2
= 1330
= 36.5 units
Use the Distance Formula to find the distance between the divers.
CONFIDENTIAL 36
1) Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula.
Now you try!
a)b)
CONFIDENTIAL 37
Now you try!
2a. (0, 9, 5) and (6, 0, 12)
2b. (5, 8, 16) and (12, 16, 20)
Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round
to the nearest tenth, if necessary.
CONFIDENTIAL 38
Surface Area of Prisms and Cylinders
Prisms and cylinders have 2 congruent parallel bases. A lateral face is not a base. The edges of the base are called base edges. A lateral edge is not an edges of a base. The lateral faces of a right prism are all rectangles. An oblique
prism has at least one nonrectangular lateral face.
Bases
Base edges
Bases
lateral faces
lateral edge
Next Page:
CONFIDENTIAL 39
An altitude of a prism or cylinder is a perpendicular segment joining the planes of the bases. The height of a three-
dimensional figure is the length of an altitude.
Surface area is the total area of all faces and curved surfaces of a three-dimensional figure. The lateral area of a prism is
the sum of the areas of the lateral faces.
altitude
Next Page:
CONFIDENTIAL 40
Lateral Area and Surface Area of Right Prisms
The lateral area of a right prism with base perimeter P and height h is L = ph.
The surface area of a right prism with lateral area L and base area B is S = L + 2B, or S = Ph + 2B.
The surface area of a cube with edge length s is S = 6s .
2
h
s
s
s
CONFIDENTIAL 41
The surface area of a right rectangular prism with length l, width w, and height h can be
written asS = 2lw + 2wh + 2lh.
CONFIDENTIAL 42
Finding Lateral Areas and surface Areas of Prisms
Find the lateral area and surface area of the right prism. Round to the nearest tenth, if necessary.
12 cm
6 cm8 cm
A) The rectangular prism L = ph = (28)12 = 336 cm P = 2(8) + 2(6) = 28 cm
S = Ph + 2 B = 336 + 2(6)(8) = 432 cm
2
2
CONFIDENTIAL 43
The lateral surface of a cylinder is the curved surface that connects the two bases. The axis of a cylinder is the segment with endpoints at the centers of the bases. The axis of a right cylinder is perpendicular to its bases. The axis of an oblique
cylinder is not perpendicular to its bases. The altitude of a right cylinder is the same length as the axis.
Bases
Bases
Axis
Axis
lateral surfaces
oblique cylinder right cylinder
CONFIDENTIAL 44
Lateral Area and Surface Area of Right Cylinders
The lateral area of a right cylinder with radius r and height h is L = 2 rh.
The surface area of a right cylinder with lateral area L and base area B is S = L + 2B, orS = 2 rh + 2 r .2
r
h
r
h
2 r
CONFIDENTIAL 45
Finding Lateral Areas and Surface Areas of Right Cylinders
A) Find the lateral area and surface area of the right cylinder. Give your answer in terms of .
2 m
5m
the radius is half the diameter, or 1 m.L = 2rh = 2(1)(5) = 10m2
S = L + 2r2 = 10 + 2(1)2 = 12m2
Next Page:
CONFIDENTIAL 46
Finding Surface Areas of composite Three-Dimensional Figures
Find the surface area of the composite figure. Round to the nearest tenth.
4 ft
20 ft
16 ft
24 ft
The surface area of the right rectangular prism is
S = Ph + 2B = 80(20) + 2(24)(16) = 2368 ft2
CONFIDENTIAL 47
Exploring Effects of Changing dimensions
The length, width, and height of the right rectangular prism are doubled. Describe the effect on the surface
area.
3 in.
2 in.6 in.
Original dimensions:S = Ph + 2B = 16(3) + 2(6)(2) = 72 in 2
Length, width, and height doubled:S = Ph + 2B = 32(6) + 2(12)(4) = 288 in
2
Notice that 288 = 4(72). If the length, width, and height are doubled, the surface area is multiplied by 2, or 4.2
CONFIDENTIAL 48
Now you try!
1) Find the lateral area and surface area of a cube with edge length 8 cm.
CONFIDENTIAL 49
Now you try!
Use the information above to answer the following.
2) A piece of ice shaped like a 5 cm by 5 cm by 1 cm rectangular prism has approximately the same volume as
the pieces in the previous slide. Compare the surface areas. Which will melt faster?
3 cm
4 cm
2 cm
5 cm
5 cm
1 cm
CUBE 1CUBE 2
CONFIDENTIAL 50
Surface Area of Pyramids and Cones
The vertex of a pyramid is the point opposite the base of the pyramid. The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base.
regular pyramid nonregular pyramid
Slant height
Vertices
Lateral faces
Altitude
Bases
Next page:
CONFIDENTIAL 51
The lateral faces of a regular pyramid can be arranged to cover half of a rectangle with a height equal to the slant height of the pyramid. The width of the rectangle is equal to the base perimeter of the pyramid.
s
s s
sl
P = 4s
l
ssss
CONFIDENTIAL 52
Lateral and surface Area of a Lateral and surface Area of a regular pyramidregular pyramid
The lateral area of a regular pyramid with perimeter P and slant height l is L = 1/2Pl.
The surface area of a regular pyramid with lateral area L and base area B is S = L + B, or S = ½ pl + B.
l
CONFIDENTIAL 53
Finding Lateral and surface Area of Pyramids
Find the lateral area and surface area of the pyramid.a.) a regular square pyramid with base edge length 5 in. and slant height 9in.
L = ½ Pl Lateral area of a regular pyramid = ½(20)(9) = 90 in P = 4(5) = 20 in.S = ½ Pl + B Surface area of a regular pyramid = 90 + 25 = 115 in B = 5 = 25 in
2
2 2 2
Next page:
CONFIDENTIAL 54
The vertex of a cone is the point opposite the base. The axis of a cone is the segment with endpoints at the vertex and the center of the base. The axis of a right cone is perpendicular to the base. The axis of an oblique cone is not perpendicular to the base.
Slant height
right cone oblique cone
Vertices
Lateral surfaces
axis axis
Base
Next page:
CONFIDENTIAL 55
The slant height of a right cone is the distance from the vertex of a right cone to a point on the edge of the base. The altitude of a cone is a perpendicular segment from the vertex of the cone to the plane of the base.
CONFIDENTIAL 56
Lateral and Surface Area of a right cone
Lateral and Surface Area of a right cone
r
l
rl
The lateral area of a right cone with radius r and slant height l is L = rl.
The surface area of a right cone with lateral area L and base area B is S = L + B, or S = rl + r 2
CONFIDENTIAL 57
Finding Lateral Area and Surface Area of right cones
Find the lateral area and surface area of the cone. Give your answers in terms of .
A) A right cone with radius 2 m and slant height 3 m.
L = rl Lateral area of a cone = (2) (3) = 6 m2 Substitute 2 for r and 3 for l.S = rl + r2 Surface area of a cone = 6 + (2)2 = 10 m2 Substitute 2 for r and 3 for l.
CONFIDENTIAL 58
Exploring Effects of Changing Dimensions
3 cm
5 cmThe radius and slant height of the right cone tripled. Describe the effect on the surface area.
Notice that 216 = 9(24). I f the radius and slant heightare tripled, the surface area is multiplied by 32, or 9.
S = rl + r2 = (9)(15) + (9)2 = 216 cm2
S = rl + r2 = (3)(5) + (3)2 = 24 cm2
Original dimensions: Radius and slant height tripled:
CONFIDENTIAL 59
Finding Surface Area of Composite Three-Dimensional Figures
Find the surface area of the composite figure.
28 cm
90
cm
45
cm
The height of the cone is 90 - 45 = 45 cm.By the Pythagorean Theorem,l = 282 + 452 = 53 cm. the lateral area of thecoan isL = rl = (28)(53) = 1484 cm2.
Next page:
CONFIDENTIAL 60
28 cm
90
cm
45
cmThe lateral area of the cylinder is L = 2rh = 2(28)(45) =2520 cm2.
The base area is B = r2 = (28)2 = 784 cm2
S = (cone lateral area) + (cylinder lateral area) + (base area) = 2520 + 784 + 1484 = 4788 cm2
CONFIDENTIAL 61
Now you try!
1) Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10ft.
CONFIDENTIAL 62
Now you try!
16 ft
6 ft
2) Find the lateral area and surface area of the right cone.
CONFIDENTIAL 63
Volume of Prisms and Cylinders
A cube built out of 27 unit cubes has a volume of 27 cubic units.
The volume of a three-dimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill
the interior.
Next Page:
CONFIDENTIAL 64
A right prism and an oblique prism with the same base and height have the same volume
Cavalieri's principle says that if two three-dimensional figure have the same height and have the same cross-
sectional area at every level, they have the same volume.
CONFIDENTIAL 65
Volume of a Prism
h
B B
h
The volume of a prism with base area B and height h is V = Bh.
Next Page:
CONFIDENTIAL 66
s
s
s
h
w
l
The volume of a right rectangular prism with length l, width w, and height h is V = lwh.
The volume of a cube with edge length s is V = s . 3
CONFIDENTIAL 67
Finding Volumes of Prisms
Find the volume of the prism. Round to the nearest tenth, if necessary.
A).
10 cm
12 cm
8 cm
volume of a right rectangular prism Substitute 10 for l, 12 for w, and 8 for h is V = lwh.
V = lwh = (10)(12)(8) = 980 cm
3
CONFIDENTIAL 68
Marine Biology Application
The aquarium at the right is a rectangular prism. Estimate the volume of the water in the aquarium in
gallons. The density of water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds.
(Hint: 1 gallon = 0.134 ft )3
8 ft
120 ft
60 ft
Next Page:
CONFIDENTIAL 69
8 ft
120 ft
60 ft
Step:1Step:1 Find the volume of the aquarium in cubic feet.
V = lwh = (120)(60)(8) = 57,600 cm 3
Next Page:
CONFIDENTIAL 70
8 ft
120 ft
60 ft
Step:2Step:2 Use the conversion factor 1 gallon
0.134 ft3 to estimate the
volume in gallons.
57,600 ft3 1 gallon
0.134 ft3 = 429,851 gallons
1 gallon
0.134 ft3 = 1
Next Page:
CONFIDENTIAL 71
Step:3Step:3
Use the conversion factor 8.33 pounds
1 gallon to estimate the weight of the
water.
429,851 gallons 8.33 pounds
1 gallon 3,580,659 pounds
8.33 pounds
1 gallons = 1
The aquarium holds about 429,851 gallons. The water in the aquarium weight about 3,580,659 pounds
8 ft
120 ft
60 ft
CONFIDENTIAL 72
Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they
have the same volume.
CONFIDENTIAL 73
r
h
r
h
The volume of a cylinder with base area B, radius r, and height h is V = Bh, or V = r h.2
CONFIDENTIAL 74
12 cm
8 cm
Find the volume of the cylinder. Give your answers both in terms of and rounded to the nearest tenth.
A).
V = r2h Volume of a cylinder = 8 2 12 Substitute 8 for r and 12 for h. = 768 cm3 2412.7 cm3
Next Page:
Finding Volumes of Cylinders
CONFIDENTIAL 75
Exploring Effects of Changing Dimensions
The radius and height of the cylinder are multiplied by ½. Describe the effect on the volume.
6 m
12 moriginal dimensions: radius and height multiplied by1
2.
V = r2h V = r2h = (6)2(12) = (3)2(6) = 432 m3 =54 m3
Notice that 54 = 1
8(432). if the radius and height are multiplied by
1
2, the volume is multiplied by
1
2 3
, or 1
8.
CONFIDENTIAL 76
Finding Volumes of Composite Three-Dimensional Figures
Find the volume of the composite figure. Round to the nearest tenth.
5 m
8 m
9 m
6 m
The base area of the prism is
B = 1
2(6)(8) =24 m2.
The volume of the prism is V = Bh = 24(9) = 216 m3.The cylinder's diameter equals the hypotenuse of the prism's base, 10 m. So the radius is 5 m.The volume of the cylinder is V = r2h = (5)2(5) = 125m3.The total volume of the figure is the sum of the volumes.V = 216 + 125 608.7 m3
CONFIDENTIAL 77
Now you try!
1) Find the volume of a triangular prism with a height of 9 yd whose base is a right triangle with
legs 7 yd and 5 yd long.
CONFIDENTIAL 78
Now you try!
2) The length, width, and height of the prism are doubled. Describe the effect on the volume.
1.5 ft
4 ft 3 ft
CONFIDENTIAL 79
Volume of Pyramids and Cones
The volume of a pyramid is related to the volume of a prism with the same base and height. The relationship can be verified by dividing a cube into three congruent square pyramids, as shown.
Next Page:
CONFIDENTIAL 80
The square pyramids are congruent, so they have the same volume. The volume of each pyramid is one third the volume of the cube.
CONFIDENTIAL 81
Volume of a Pyramid
The volume of a pyramid with base area B and height h is V = 1/3 Bh.
h h
BB
CONFIDENTIAL 82
Finding Volumes of Pyramids
Next Page:
4 in.
4 in.
6 in.
Find the volume of each pyramid.
a)A rectangular pyramid with length 7 ft, width 9 ft, and height 12 ft.
b) The square pyramid the base is a square with a side length of 4 in., and the height is 6 in.
V = 1
3Bh =
1
3(42)(6) = 32 in3
V = 1
3Bh =
1
3(7 9)(12) = 252 ft3
CONFIDENTIAL 83
Architecture Application
The Rainforest Pyramid in Galveston, Texas, is a square pyramid with a base area of about 1 acre and a height of 10 stories. Estimate the volume in cubic yards and in cubic feet. (Hint: 1 acre = 4840 yd, 1 story = 10 ft)
2
1 acre
10 storiesThe base is a squarewith an area of about4840 yd2. the base edgelength is 4840 = 70 yd.the height is about10(10) = 100 ft, orabout 33 yd.
Next Page:
First find the volume in cubic yards.
V = 1
3Bh Volume of a regular pyramid
= 1
3(702)(33) = 53,900 yd3 Substitute 702 for B and 33 for h.
CONFIDENTIAL 84
1 acre
10 stories
Then convert your answer to find the volume in cubic feet. thevolume of one cubic yard is (3 ft)(3 ft)(3 ft) = 27 ft3.
Use the conversion factor 27 yd3
1 yd3 to find the volume in cubic feet.
53,900 yd3 27 yd3
1 yd3 1,455,300 ft3
CONFIDENTIAL 85
Volume of a Cones
h
r
h
r
The volume of a cone with base area B, radius r,
and height h is V = 1
3Bh, or V =
1
3r2h.
CONFIDENTIAL 86
Finding Volumes of a Cones
Find the volume of each cone. Give your answers both in terms of and rounded to the nearest tenth.
A) A cone with radius 5 cm and height 12 cm
Next Page:
V =1
3r2h Volume of a cone
=1
3(5)2 (12) Substitute 5 for r and 12 for h.
= 100 cm3 314.2 cm3 Simplify.
CONFIDENTIAL 87
B) A cone with a base circumference of 21 cm and a height 3 cm less than twice the radius
Step 1: Use the circumference to find the radius.
Step 2: Use the radius to find the height.
2(10.5) – 3 = 18 cm The height is 3 cm less than twice the radius.
Step 3: Use the radius and height to find the volume.
2r = 21 Substitute 21 for C. r = 10.5 cm Divide both sides by 2.
V = 1
3 r2h Volume of a cone
= 1
3(10.5)2 (18) Substitute 10.5 for r and 18 for h.
=661.5 cm3 2078.2 cm3 Simplify.Next Page:
CONFIDENTIAL 88
25 ft
7 ft
Step 1: Use the Pythagorean Theorem to find the height.
Step 2: Use the radius and height to find the volume.
V = 1
3 r2h Volume of a cone
= 1
3(7)2 (24) Substitute 7 for r and 24 for h.
=392 ft3 1231.5 ft3 Simplify.
72 + h2 = 252 Pythgorean Theorem h2 = 576 Subtract 72 from both sides. h = 24 Take the square root of both sides.
CONFIDENTIAL 89
Exploring Effects of Changing Dimensions
The length, width, and height of the rectangular pyramid are multiplied by ¼ . Describe the effect on the volume.
20 ft
24 ft20 ft
Next Page:
CONFIDENTIAL 90
20 ft
24 ft20 ft
Length, width, and height multiplied by ¼:
Original dimensions:
V = 1
3Bh
= 1
3(24 20)(20)
= 3200 ft3
V = 1
3Bh
= 1
3(6 5)(5)
= 50 ft3
Notice that 50 = 1
64(3200). I f the length, width, and height
are multiplied by 1
4, the volume is multiplied by
1
4 3
, or 1
64.
CONFIDENTIAL 91
Finding Volumes of Composite Three-Dimensional Figures
Find the volume of the composite figure. Round to the nearest tenth.
2 in
4 in
5 in
The volume of the cylinder is V = r2h = 2 2 (2) = 8 in3.The volume of the cone is
V= 1
3r2h =
1
3 2 2(3) = 4 in3.
The volume of the composite figure isthe sum of the volumes. V= 8 + 4 = 12 in3 37.7 in3
CONFIDENTIAL 92
Now you try!
1) Find the volume of a regular hexagonal pyramid with a base edge length of 2 cm and a height equal to the area of
the base.
CONFIDENTIAL 93
Now you try!
2) The radius and height of the cone are doubled. Describe the effect on the volume.
9 cm
18 cm
CONFIDENTIAL 94
Spheres
A sphere is the locus of points in space that are a fixed distance from a given point called the center of a sphere. A radius of a sphere connects the center of the sphere to any point on the sphere to any point on the sphere. A hemisphere is half of a sphere. A great circle divides a sphere into two hemispheres.
Center
HemisphereGreat circle
Radius
Next Page:
CONFIDENTIAL 95
The figure shows a hemisphere and a cylinder with a cone removed from its interior. The cross sections have the same area at every level, so the volumes are equal by Cavalieri’s Principle.
h
r
Next Page:
h
r
CONFIDENTIAL 96
h
r
V(hemisphere) = V(cylinder) - V(cone)
= r2h - 1
3 r2h
= 2
3 r2h
= 2
3 r2(r) The height of the hemisphere is equal
to the radius.
= 2
3 r3
The volume of a sphere with radius r is twice the volume of
the hemisphere, or V = 4
3 r3.
h
r
CONFIDENTIAL 97
Volume of a Sphere
r
The volume of a sphere with radius r is V = 4
3 r3.
CONFIDENTIAL 98
Finding Volumes of Spheres
9 cm
Find each measurement. Give your answer in terms of .
A) The volume of the sphere
V = 4
3 r3
V = 4
3 (9)2 Substitute 9 for r.
= 972 cm2 Simplify.
Next Page:
CONFIDENTIAL 99
9 cm
972 = 4
3 r3 Substitute 972 for V.
729 = r3 Divide both sides by 4
3.
r = 9 Take the cube root of both sides.
d = 18 in. d = 2r
B) The diameter of a sphere with volume 972 in 3
Next Page:
CONFIDENTIAL 100
4 m
C) The volume of the hemisphere
V = 2
3 r3 Volume of a hemisphere
= 2
3 4 3 =
128
3 m3 Substitute 4 for r.
CONFIDENTIAL 101
In the figure, the vertex of the pyramid is at the center of the sphere. The height of the pyramid is approximate the radius r of the sphere. Suppose the entire sphere is filled with n pyramids that each have base area B and height r.
Next Page:
CONFIDENTIAL 102
V(sphere) 1
3Br +
1
3Br + ......+
1
3Br The sphere's volume is close to the
sum of the volumes of the pyramids.
4
3r3 n
1
3Br
4r2 nB Divide both sides by 1
3r.
Next Page:
CONFIDENTIAL 103
If the pyramids fill the sphere, the total area of the bases is approximate equal to the surface area of the sphere S, so4 r = S. As the number of pyramids increases, the approximation gets closer to the actual surface area.
2
CONFIDENTIAL 104
r
Surface Area of a Sphere
The surface area of a sphere with radius r is S = 4 r .2
CONFIDENTIAL 105
Finding Surface Area of Spheres
Find each measurement. Give your answers in terms of .
A) the surface area of a sphere with diameter 10 ft.
S = 4r2
S = 4(5)2 = 200 ft2 Substitute 5 for r.
B) the volume of a sphere with surface area 144 m2
S = 4r2
144 = 4r2 Substitute 144 for S. 6 = r Solve for r.
V = 4
3r3
= 4
3 6 3 = 288 m3 Substitute 6 for r.
The volume of the sphere is 288 m3.
CONFIDENTIAL 106
Now you try!
50 cm
3) Find the surface area of the sphere.
CONFIDENTIAL 107
Exploring Effects of Changing Dimensions
The radius of the sphere is tripled. Describe the effect on the volume.
3 m
Original dimensions: radius tripled:
V = 4
3r3 V =
4
3r3
= 4
3 3 3 =
4
3 9 3
= 36 m3 = 972 m3
Notice that 972 = 27(36). I f the radius is tripled,the volume is multiplied by 27.
CONFIDENTIAL 108
Finding Surface Areas and Volumes of Composite Figures
Find the surface area and volume of the composite figure. Give your answers in terms of .
7 cm
25 cm
Step 1 Find the surface area of the composite figure. The surface area of the composite figure is the sum of the surface area of the hemisphere and the lateral area of the cone.
Next Page:
S(hemisphere) = 1
2(4r2) = 2 7 2 = 98 cm2
L(cone) = rl = (7)(25) = 175 cm2
The surface area of the composite figure is 98 + 175 = 273 cm2.
CONFIDENTIAL 109
7 cm
25 cm
Step 2 Find the volume of the composite figure. First find the height of the cone.
h = 252 - 72 Pythagorean Theorem = 576 = 24 cm Simplify.
The volume of the composite figure is the sum of the volume of the hemisphere and the volume of the cone.
V(hemiphere) = 1
2
4
3r3 =
2
3 7 3 =
686
3 cm3
V(cone) = 1
3r2h =
1
3 7 2 24 = 392 cm3
The volume of the composite figureis 686
3 + 392 =
1862
3 cm3.
CONFIDENTIAL 110
Now you try!
1) Find the radius of a sphere with volume 2304 ft .3
CONFIDENTIAL 111
Now you try!
3 ft
5 ft
2) Find the surface area and volume of the composite figure.
CONFIDENTIAL 112
You did a You did a greatgreat job job today!today!