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GT/Honors Geometry
April 6 to April 15, 2015
Topic
Date
Monday
3/30
Tuesday
3/31
Wednesday
4/1
Thursday
4/2
Friday
4/3
Homework
ELA I STAAR TEST
ELA I STAAR TEST
Test Review – Surface Areas, Nets,
Cross Sections, Drawings
Test Review
ELA II STAAR TEST
ELA II STAAR TEST
TEST: SURFACE AREA
EASTER HOLIDAY
Volume of revolutions
EASTER HOLIDAY
Monday 4/6
11-4 Volumes of
Prisms & Cylinders
Page 627: (1, 2, 4-6, 10-12, 15, 17-19, 22, 26-35, 37
(part a only), 38)
Tuesday
4/7
11-5 Volumes of Pyramids & Cones
Page 634 (1, 2, 6, 8, 9, 11, 13-16, 18, 20-25, 27, 2931)
Wednesday
4/8
Thursday
4/9
Friday
4/10
11-6 Surface Area & Volume of a Sphere
Page 640 (1, 2, 5, 8, 10, 12, 13, 16-24, 29, 34, 35,
37, 38)
11.7 Areas & Volumes of Similar Solids
Quiz: 11-5, 11-6
Page 648-649 (1-4, 7-9, 12, 13, 14, 17, 18)
MC Worksheet: Areas and Volume of similar solids
Page 642 (40-43, 47)
Page 644 (10)
Worksheet-Area and Volume of Composite Figures
Monday 4/13
Frustums
Tuesday
4/14
Wednesday
4/15
Review for test
Page 636 (32, 34)
Page 649 (11, 16, 37)
Page 652 (1-2)
Worksheet – Fun with Frustums
Worksheet : Surface Area and Volume review
Test: Surface Area and Volume
TBA
Volume of rotation and composite solids
Name__________________________________
Date____________________Period__________
H/W FOR Solids Formed by Rotation
Graph of Region
Describe and draw the solid formed
when the figure is rotated about
y-axis
Describe and draw the solid formed
when the figure is rotated about
x-axis
y
y
x
y
x
y
x
y
x
y
x
y
x
y
x
x
Solids Formed by Rotation
Graph of Region
Describe and draw the solid formed
when the figure is rotated about
y-axis
Describe and draw the solid formed
when the figure is rotated about
x-axis
y
y
x
y
x
y
x
y
x
y
x
y
x
y
x
x
NOTE. 11-4 Volume of Prisms and Cylinders
Volume is the space that a three-dimensional figure occupies. Since it is 3-D, it is measured in cubic
units.
Cavalieri’s Principle – If two 3-D figures have the same height and the same cross-sectional area at every
level, then they have the same volume.
The following prisms have the same height. Since the area of each cross section is 6 in2, by Cavalieri’s
Principle, their volumes will be the same.
3 in
2 in
Base is a 3 in by
2 in rectangle
Base is a
parallelogram
2 in
3 in
The volume of a prism or cylinder is the product of the base area and the height.
Find the volume of each of the following prisms or cylinders. Give exact answers and answers rounded to
the nearest tenth.
1.
2.
20 ft
8 cm
20 ft
12 cm
32 ft
24 ft
3.
5
20
4.
10
13
6
15
5. The volume of a cylinder with height 8 in is 200 π in3. Find the length of the radius.
7
6. Find the volume of the right prism. (note: all angles are right angles.
21
25
5
30
y
7. The plane region is revolved completely about the y-axis.
Describe the sold and find its volume in terms of π.
x
8. A cylindrical “hole” (with diameter 6 cm) has been cut out of a prism. Find the volume of the remaining
solid.
15 cm
12 cm
22 cm
11-5 Volume of Pyramids and Cones
The volume of a pyramid is one third the product of the area of the base and the height of the pyramid.
V =
1
Bh
3
Because of Cavalieri’s Principle, the volume formula is true for all pyramids, including oblique pyramids.
The height of an oblique pyramid is the length of the perpendicular segment from the vertex to the plane
of the base.
The volume of a cone is one third the product of the area of the base and the height of the cone.
V =
1
1
Bh, or V =
πr2
3
3
This formula applies to all cones, including oblique cones.
Examples: Find the volume of each of the following solids.
1.
3.
5.
10 cm
2.
15 ft
4.
6.
7. Find the radius of a circular cone whose volume
is 8π in3 and height is 6 in.
2 ft
6 ft
8. Find x if the volume is 126 cm2.
9. Find the volume of a circular cone whose radius
is 12 ft and whose surface area is 300π ft2.
14 cm
9 cm
x
10. A 120˚ sector is cut from a circle with a radius of 9 in. The 120˚ sector is “rolled” to create a cone.
Find the volume of the resulting cone.
120˚
9 in
11. The plane region is revolved completely about the given line to sweep out a solid of revolution.
y
Describe
the solid and then find its volume in terms of π.
(a) about the x-axis
(b) about the y-axis
x
11-6 Surface Areas and Volumes of Spheres –
A _______________ is the set of all points in space equidistant from a given point called the
_________________. A ________________ is a segment that has one endpoint at the
center and the other endpoint on the sphere. A __________________ is a segment passing
through the center with endpoints on the sphere.
When a plane and a sphere intersect in more than one point, the intersection is a ___________.
If the center of the circle is also the center of the sphere, the circle is called a ___________
_____________. The circumference of a great circle is the ___________________ of the
sphere. A great circle divides a sphere into two _____________________.
S = 4πr2
Surface Area of a Sphere:
Volume of a Sphere:
V =
4
3
r3
Example 1: Find the surface area of a sphere whose diameter is 18 ft.
Example 2: Find the surface area of a sphere where one of its great circles has an area of 36π in2.
Example 3: The circumference of a rubber ball is 13 cm. Find its surface area to the nearest
whole number.
Example 4: Find the volume of the sphere whose diameter is 30 cm.
Example 5: The volume of a sphere is
32
3
π m3. Find the surface area of the sphere in terms of π.
Example 6: The volume of a sphere is 7238 in3. Find its surface area to the nearest whole
number.
S●
●C
●R
Example 7: C is the center of the sphere.
Plane B intersects the sphere in circle R.
(a) Suppose CR = 5 and SR = 12. What is the length of a radius of the sphere?
(b) If the radius of the sphere is 41 and the radius of circle R is 40, find CR.
B
11-7 Study Guide – Areas and Volumes of Similar Solids
Solids that have the same shape but different in size are said to be similar. You can tell if two solids are similar
by comparing the ratios of corresponding linear measurements.
Determine if the two solids are similar. If so, give the similarity ratio.
1.
2.
3.
The following two cylinders are similar. Fill in the table below.
radius
Circumference
height
Base area
Lateral
Volume
4
3
Area
Big
12
9
Little
Linear ratios (similarity ratio)
Area ratios
Volume ratios
Given that these two cones are similar.
4) Find the similarity ratio.
5) Find the ratio of their diameters.
5
3
6) What is the ratio of their base areas?
7) What is the ratio of their volumes?
8) If the lateral area of the little cone is 60 in2, find the lateral area of the big cone.
9) If the volume of the big cone is 600 cm3, find the volume of the little cone.
10) The ratio of the slant height of two pyramids is 2 to 5 and the surface area of the larger pyramid is 105 cm2. Find the
surface area of the smaller pyramid.
11) Two similar prisms have surface areas in a ratio of 9 to 16. If the volume of the smaller prism is 67.5 in 3, find the
volume of the larger prism.
1
Mr. Baskin has two conical cups. The diameter of the first cup is the same as the diameter of the second cup. The height of
the first cup is half the height of the second cup. Compare the volume of the second cup to the volume of the first cup.
The volume is the same for both cups
The volume of the second cup is double the volume of the first cup.
The volume of the second cup is half the volume of the first cup.
The volume of the second cup is quadruple the volume of the first cup.
2
What is the volume of a similar rectangular box with dimensions that are 4.5 times larger than the dimensions of the
rectangular box shown below?
15,552 in.3
69,984 in.3
139,968 in.3
314,928 in.3
3
If the area of an equilateral triangle is increased by a factor of 9, what is the change in the length of the sides of the
equilateral triangle?
The length is 3 times the original length.
The length is 4.5 times the original length.
The length is 9 times the original length.
The length is 18 times the original length.
4
Use test tools
If the surface area of a cube is increased by a factor of 16, what is the change in the length of each side of the cube?
The length is 2 times the original length.
The length is 4 times the original length.
The length is 8 times the original length.
The length is 16 times the original length.
Use test tools
A cylindrical column in a building has a volume of 114 cubic feet. In another part of the building, another cylindrical column
5
has the same base area, but three times the height. What is the volume of the taller column?
38 ft3
342 ft3
1026 ft3
3078 ft3
Use test tools
6
A square pyramid has a volume of 108 cubic meters. A similar second pyramid has edges and height that are
those of
the original pyramid. What is the volume of the second pyramid?
3 m3
4 m3
12 m3
36 m3
7
Use test tools
A rectangular solid has volume of 24 cubic decimeters. If the length, width, and height are all changed to
their original
size, what will be the new volume of the rectangular solid?
8
Use test tools
If the surface area of a cube is increased by a factor of 4, what is the change in the length of the sides of the cube?
The length is 2 times the original length.
The length is 4 times the original length.
The length is 6 times the original length.
The length is 8 times the original length.
9
Use test tools
Mr. Kelly\'s company manufactures a cylindrical soup can that has a diameter of 6 inches and a volume of 226 cubic inches.
If the diameter stays the same and the height is doubled, what will happen to the can\'s volume?
It will remain the same.
It will double.
It will triple.
It will quadruple.
Use test tools
10
The rectangle below has a perimeter of 18 feet with a length of 6 feet.
A new rectangle is formed by decreasing the width of the original rectangle by 1 foot and keeping the length the same. How
will the perimeter of the new rectangle compare with the perimeter of the original rectangle?
The perimeter of the new rectangle will be 3 feet shorter than the perimeter of the original rectangle.
The perimeter of the new rectangle will be 2 feet shorter than the perimeter of the original rectangle.
The perimeter of the new rectangle will be 1 foot shorter than the perimeter of the original rectangle.
The perimeter of the new rectangle will be
11
foot shorter than the perimeter of the original rectangle.
Use test tools
If the height and radius of a traffic cone are both divided by 3, what is the effect on its surface area?
The surface area is 3 times the original surface area
The surface area is 9 times the original surface area
12
The surface area is
times the original surface area
The surface area is
times the original surface area
Use test tools
A company packages their product in two sizes of cylinders. Each dimension of the larger cylinder is twice the
size of the corresponding dimension of the smaller cylinder.
Based on this information, which of the following statements is true?
The volume of the larger cylinder is 2 times the volume of the smaller cylinder.
The volume of the larger cylinder is 4 times the volume of the smaller cylinder.
The volume of the larger cylinder is 8 times the volume of the smaller cylinder.
The volume of the larger cylinder is 6 times the volume of the smaller cylinder.
Ws Homework 4/10. Surface area and volume of composite solids
Find the Surface area and volume of the following solids
1.
The solid is made up of a square prism
and a square pyramid
2. Find SA and Volume of the solid
3. Find SA and Volume of the solid
4. A cone is carved out of the cylinder. Find SA and Volume of the remaining solid
5. A cylindrical hole is drilled in the cylinder. Find SA and Volume of the remaining solid.
6. A rectangular prism is removed from the solid as shown.
Find SA and Volume of the remaining solid
4/13. Notes for Frustums
A frustum of a cone is the part that remains when the
vertex is cut off by a plane parallel to its base.
A frustum of a pyramid is the part that remains when
the vertex is cut off by a plane parallel to its base.
Original is Square-based Pyramid
26 in,.
? cm.
24 cm.
12 cm
10 in.
12 cm
20 in.
18 cm
? cm
18 cm
1. How is the cut-off cone related to the original cone?
___________________________________________
1. How is the cut-off pyramid related to the original
pyramid? _________________________________
2. What is the similarity ratio for the two cones? ____
2. What is the similarity ratio for the two pyramids?
_________
3. Use the similarity ratio to find the height of the
cutoff cone. ___________
3. What is the ratio of the volume of the cut-off
pyramid to the original pyramid? _______________
4. What is the height of the frustum?
______________
4. If the volume of the original pyramid is 256 cm.3,
find the volume of the cut-off pyramid. __________
What is the volume of the frustum? ___________
5. Can you find the surface area of the frustum by
subtracting the surface area of the cut-off cone from the
surface area of the original cone? ______________
Why or why not? _______________________
5. What is the ratio of the volume of the frustum to
the original pyramid? _____________
6. The ratios in (3) and (5) must add up to ________.
6. Can you find the volume of the frustum by
subtracting the volume of the cut-off cone from the
volume of the original cone? _________________
7. Find the volume of the frustum to the nearest tenth.
7. What shape are the lateral faces of the frustum?
_________________
8. If the lateral edge of the original pyramid is 26 in.,
find the height of the trapezoidal lateral faces.
__________. Find the LA for the frustum. ________
8. If you cut along the slant height of the original cone
and then unwrap the lateral area of the frustum, what
shape will it be? ________________ Find the lateral
area of the frustrum. ___________________
9. Find the surface area of the frustum. ___________
Example 1
Find the exact volume and find the surface area to the nearest whole number.
3 cm
10 cm
8 cm
Example 2
Find the exact volume and the surface area to the nearest whole number.
8 cm.
9 cm.
12 cm.
Example 3 - A frustum is formed by slicing a cone
parallel to its base. The ratio of the heights of the
cutoff cone to the original cone is 3:5. If the volume
of the original cone is 625 cu. in., what is the volume
of the frustum?
Example 4 – A frustum is formed by slicing a
square-based pyramid parallel to its base and
bisecting the height of the pyramid. If the lateral
area of the original pyramid is 400 sq. in., what is
the lateral area of the frustum?
Worksheet h/w for 4/13: FUN with Frustums!
1. Find the surface area of the frustum:
12 in.
5 in.
15 in.
2. The following trapezoid is rotated about the y-axis. Find the surface area of the solid generated.
3 cm
60°
6 cm
3. Find the surface area and volume of the frustum below:
9 cm
7 cm
12 cm
4. At Oaklawn Stables a watering tank for horses has the shape of a frustum of a regular square
pyramid. A bottom edge is 6 feet long and a top edge 8 feet long. If the tank has a depth of 2 feet,
how many gallons of water does it hold when filled to capacity? (1 cu. ft = 7.5 gallons)
8
2
6
5. A brass candle holder is in the shape of a frustum of a right circular cone with bottom diameter 6
cm., top diameter 4.5 cm., and height 3.5 cm. The candle-insert hole is in the shape of a cylinder
with diameter 2.5 cm. and height 3 cm. If the weight of the brass used is 8.7 grams per cubic
centimeter, find the weight of the candleholder.
Test Review : Volume and Area
1. A right triangle, 5 in. x 12 in. x 13 in. is revolved to create a solid. Find the volume if it is
revolved about the short leg. Find the volume if it is revolved about the long leg. Which one
has the greater volume?
2. The two water pipes of same length have inside diameters of 6 cm and 8 cm. These two
pipes are replaced by a single pipe of the same length which has the same capacity as the
smaller pipes combined. What is the inside diameter of the new pipe?
3. A landscape design specifies that topsoil to a depth of 4 in. be spread over a field that is 76
yd by 32 yd. How many cubic yards of topsoil should be ordered?
4. A 4x7 rectangle can be rotated about the long side to generate a cylinder. It also can be
rotated about the short side to generate a cylinder. Find the ratio of the volumes of these
cylinders.
5. Which holds more, a cone-shaped drinking cup with 6 in base diameter and 6 in height or a
cylindrical drinking cup 4 in diameter base and 4 in height?
6. How long will it take to fill the pool with a garden hose that has a flow rate of 5 gal/min?
Note: 7.5 gallons = 1 ft3
20 ft
3 ft
10 ft
8 ft
13 ft
10 ft