12-6 Surface Area and Volumes of Spheres
Find the surface area of each sphere or hemisphere. Round to the nearest tenth.
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sphere: area of great circle = 36ʌ yd
2
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We know that the area of a great circle is
. Find r.
Now find the surface area.
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hemisphere: circumference of great circle § 26 cm
Page 1
12-6 Surface Area and Volumes of Spheres
sphere: area of great circle = 36ʌ yd
2
62/87,21
We know that the area of a great circle is
. Find r.
Now find the surface area.
hemisphere: circumference of great circle § 26 cm
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We know that the circumference of a great circle is
.
The area of a hemisphere is one-half the area of the sphere plus the area of the great circle.
Find the volume of each sphere or hemisphere. Round to the nearest tenth.
sphere: radius = 10 ft
62/87,21
hemisphere: diameter = 16 cm
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62/87,21
Page 2
12-6 Surface Area and Volumes of Spheres
hemisphere: diameter = 16 cm
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hemisphere: circumference of great circle = 24ʌ m
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We know that the circumference of a great circle is
. Find r.
Now find the volume.
2
sphere: area of great circle = 55ʌ in
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We know that the area of a great circle is
. Find r.
Now find the volume.
BASKETBALL Basketballs used in professional games must have a circumference of 29
surface area of a basketball used in a professional game?
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62/87,21
We know that the circumference of a great circle is 2ʌr . Find r.
inches. What is the
Page 3
12-6 Surface Area and Volumes of Spheres
BASKETBALL Basketballs used in professional games must have a circumference of 29
inches. What is the
surface area of a basketball used in a professional game?
62/87,21
We know that the circumference of a great circle is 2ʌr . Find r.
Find the surface area.
Find the surface area of each sphere or hemisphere. Round to the nearest tenth.
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62/87,21
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Page 4
12-6Surface Area and Volumes of Spheres
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62/87,21
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sphere: circumference of great circle = 2ʌ cm
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We know
the circumference
of a great circle is 2ʌr. Find r.
eSolutions
Manual that
- Powered
by Cognero
Page 5
12-6Surface Area and Volumes of Spheres
sphere: circumference of great circle = 2ʌ cm
62/87,21
We know that the circumference of a great circle is 2ʌr. Find r.
sphere: area of great circle § 32 ft
2
62/87,21
2
We know that the area of a great circle is ʌr . Find r.
2
hemisphere: area of great circle § 40 in
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2
We know that the area of a great circle is ʌr . Find r.
Substitute for r2 in the surface area formula.
hemisphere: circumference of great circle = 15ʌ mm
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62/87,21
Page 6
12-6 Surface Area and Volumes of Spheres
hemisphere: circumference of great circle = 15ʌ mm
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Find the volume of each sphere or hemisphere. Round to the nearest tenth.
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sphere: radius = 1.4 yd
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Page 7
12-6 Surface Area and Volumes of Spheres
sphere: radius = 1.4 yd
62/87,21
hemisphere: diameter = 21.8 cm
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2
sphere: area of great circle = 49ʌ m
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Find r.
Find the volume.
sphere: circumference of great circle § 22 in.
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Find r.
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eSolutions
Find the volume.
Page 8
12-6 Surface Area and Volumes of Spheres
sphere: circumference of great circle § 22 in.
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Find r.
Find the volume.
hemisphere: circumference of great circle § 18 ft
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Find r.
Find the volume.
hemisphere: area of great circle § 35 m
2
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Find r.
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eSolutions
Page 9
12-6 Surface Area and Volumes of Spheres
hemisphere: area of great circle § 35 m
2
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Find r.
Find the volume.
FISH A puffer fish is able to ³puff up´when threatened by gulping water and inflating its body. The puffer fish at
the right is approximately a sphere with a diameter of 5 inches. Its surface area when inflated is about 1.5 times its
normal surface area. What is the surface area of the fish when it is not puffed up?
5HIHUWRWKHSKRWRRQ3DJH
62/87,21
Find the surface area of the non-puffed up fish, or when it is normal.
ARCHITECTURE The Reunion Tower in Dallas, Texas, is topped by a spherical dome that has a surface area of
approximately 13,924ʌ square feet. What is the volume of the dome? Round to the nearest tenth.
5HIHUWRWKHSKRWRRQ3DJH
62/87,21
Find r.
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Page 10
12-6Surface Area and Volumes of Spheres
ARCHITECTURE The Reunion Tower in Dallas, Texas, is topped by a spherical dome that has a surface area of
approximately 13,924ʌ square feet. What is the volume of the dome? Round to the nearest tenth.
5HIHUWRWKHSKRWRRQ3DJH
62/87,21
Find r.
Find the volume.
TREE HOUSE The spherical tree house, or tree sphere, has a diameter of 10.5 feet. Its volume is 1.8 times the
volume of the first tree sphere that was built. What was the diameter of the first tree sphere? Round to the nearest
foot.
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The volume of the spherical tree house is 1.8 times the volume of the first tree sphere.
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The diameter is 4.3(2) or about 9 ft.
Find the surface area and the volume of each solid. Round to the nearest tenth.
Page 11
12-6 Surface Area and Volumes of Spheres
TREE HOUSE The spherical tree house, or tree sphere, has a diameter of 10.5 feet. Its volume is 1.8 times the
volume of the first tree sphere that was built. What was the diameter of the first tree sphere? Round to the nearest
foot.
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The volume of the spherical tree house is 1.8 times the volume of the first tree sphere.
The diameter is 4.3(2) or about 9 ft.
Find the surface area and the volume of each solid. Round to the nearest tenth.
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To find the surface area of the figure, calculate the surface area of the cylinder (without the bases), hemisphere
(without the base), and the base, and add them.
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To find the volume of the figure, calculate the volume of the cylinder and the hemisphere and add them.
Page 12
12-6Surface Area and Volumes of Spheres
The diameter is 4.3(2) or about 9 ft.
Find the surface area and the volume of each solid. Round to the nearest tenth.
62/87,21
To find the surface area of the figure, calculate the surface area of the cylinder (without the bases), hemisphere
(without the base), and the base, and add them.
To find the volume of the figure, calculate the volume of the cylinder and the hemisphere and add them.
62/87,21
To find the surface area of the figure, we will need to calculate the surface area of the prism, the area of the top
base,Manual
the area
of thebygreat
circle, and one-KDOIWKHVXUIDFHDUHDRIWKHVSKHUH
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Page 13
)LUVWILQGWKHVXUIDFHDUHDRIWKHSULVPDQGVXEWUDFWWKHDUHDRIWKHWRSEDVH
12-662/87,21
Surface Area and Volumes of Spheres
To find the surface area of the figure, we will need to calculate the surface area of the prism, the area of the top
base, the area of the great circle, and one-KDOIWKHVXUIDFHDUHDRIWKHVSKHUH
)LUVWILQGWKHVXUIDFHDUHDRIWKHSULVPDQGVXEWUDFWWKHDUHDRIWKHWRSEDVH
This covers everything but the top of the figure. The corners of the top base can be found by subtracting the area of
the great circle from the area of the base.
Next, find the surface area of the hemisphere.
Now, find the total surface area.
The volume of the figure is the volume of the prism minus the volume of the hemisphere.
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Page 14
12-6 Surface Area and Volumes of Spheres
TOYS The spinning top is a composite of a cone and a hemisphere.
a. Find the surface area and the volume of the top. Round to the nearest tenth.
b. If the manufacturer of the top makes another model with dimensions that are one-half of the dimensions of this
top, what are its surface area and volume?
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a. Use the Pythagorean Theorem to find the slant height.
Lateral area of the cone:
Surface area of the hemisphere:
b. Use the Pythagorean Theorem to find the slant height.
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Page 15
12-6 Surface Area and Volumes of Spheres
b. Use the Pythagorean Theorem to find the slant height.
Lateral area of the cone:
Surface area of the hemisphere:
BALLOONS A spherical helium-filled balloon with a diameter of 30 centimeters can lift a 14-gram object. Find the
size of a balloon that could lift a person who weighs 65 kilograms. Round to the nearest tenth.
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1 kg = 1000 g.
Form a proportion.
Let x be the unknown.
The balloon would have to have a diameter of approximately 139,286 cm.
Use sphere S to name each of the following.
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Page 16
12-6 Surface Area and Volumes of Spheres
The balloon would have to have a diameter of approximately 139,286 cm.
Use sphere S to name each of the following.
a chord
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A chord of a sphere is a segment that connects any two points on the sphere.
a radius
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A radius of a sphere is a segment from the center to a point on the sphere.
Sample answer:
a diameter
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A diameter of a sphere is a chord that contains the center.
a tangent
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A tangent to a sphere is a line that intersects the sphere in exactly one point. Line l intersects the sphere at A .
a great circle
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If the circle contains the center of the sphere, the intersection is called a great circle.
S
DIMENSIONAL ANALYSIS Which has greater volume: a sphere with a radius of 2.3 yards or a cylinder with a
radius of 4 feet and height of 8 feet?
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2.3 yards
2.3(3) or
feet
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Cognero
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If the circle contains the center of the sphere, the intersection is called a great circle.
12-6 Surface Area and Volumes of Spheres
S
DIMENSIONAL ANALYSIS Which has greater volume: a sphere with a radius of 2.3 yards or a cylinder with a
radius of 4 feet and height of 8 feet?
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2.3 yards = 2.3(3) or 6.9 feet
3
3
Since 1376 ft > 402 ft , the sphere has the greater volume.
FOOD Suppose the orange in the graphic is a sphere with a radius of 4 centimeters. Round to the nearest tenth.
a. What is the volume of the portion of the sphere that represents orange production in California?
b. What is the surface area of the portion of the sphere that represents orange production in Florida?
62/87,21
a. Find the volume of the sphere with radius 4 cm.
b. Find the surface area of the sphere with radius 4 cm.
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Page 18
12-6 Surface Area and Volumes of Spheres
3
3
Since 1376 ft > 402 ft , the sphere has the greater volume.
FOOD Suppose the orange in the graphic is a sphere with a radius of 4 centimeters. Round to the nearest tenth.
a. What is the volume of the portion of the sphere that represents orange production in California?
b. What is the surface area of the portion of the sphere that represents orange production in Florida?
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a. Find the volume of the sphere with radius 4 cm.
b. Find the surface area of the sphere with radius 4 cm.
Describe the number and types of planes that produce reflection symmetry in each solid. Then describe
the angles of rotation that produce rotation symmetry in each solid.
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Page 19
62/87,21
7KHUHLVDQLQILQLWHQXPEHURISODQHVWKDWSURGXFHUHIOHFWLRQV\PPHWU\IRUVSKHUHV$Q\SODQHWKDWSDVVHVWKURXJKWK
12-6 Surface Area and Volumes of Spheres
Describe the number and types of planes that produce reflection symmetry in each solid. Then describe
the angles of rotation that produce rotation symmetry in each solid.
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7KHUHLVDQLQILQLWHQXPEHURISODQHVWKDWSURGXFHUHIOHFWLRQV\PPHWU\IRUVSKHUHV$Q\SODQHWKDWSDVVHVWKURXJKWK
symmetry. Any plane that does not pass through the center will not produce symmetry.
These are all symmetric
These are not:
The angle of rotation is the angle through which a preimage is rotated to form the image. A sphere can be rotated at
will be identical to the preimage.
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Page 20
12-6 Surface Area and
Volumes of Spheres
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$Q\SODQHRIV\PPHWU\PXVWSDVVWKURXJKWKHRULJLQ7DNHDORRNDWDIHZH[DPSOHV
symmetric
Symmetric
Not symmetric
Not symmetric
:HVHHWKDWWKHRQO\SODQHVWKDWSURGXFHUHIOHFWLYHV\PPHWU\DUHYHUWLFDOSODQHVWKDWSDVVWKURXJKWKHRULJLQ
The angle of rotation is the angle through which a preimage is rotated to form the image. A hemisphere can be
rotated at any angle around the vertical axis, and the resulting image will be identical to the preimage. A hemisphere
URWDWHGDERXWDQ\RWKHUD[LVZLOOQRWEHV\PPHWULF
eSolutions Manual - Powered by Cognero
Page 21
:HVHHWKDWWKHRQO\SODQHVWKDWSURGXFHUHIOHFWLYHV\PPHWU\DUHYHUWLFDOSODQHVWKDWSDVVWKURXJKWKHRULJLQ
angleArea
of rotation
is the angle
12-6The
Surface
and Volumes
of through
Sphereswhich a preimage is rotated to form the image. A hemisphere can be
rotated at any angle around the vertical axis, and the resulting image will be identical to the preimage. A hemisphere
URWDWHGDERXWDQ\RWKHUD[LVZLOOQRWEHV\PPHWULF
Symmetric
Not Symmetric
CHANGING DIMENSIONS A sphere has a radius of 12 centimeters. Describe how each change affects
the surface area and the volume of the sphere.
The radius is multiplied by 4.
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· 2
3
The surface area is multiplied by 4 or 16. The volume is multiplied by 4 or 64.
The radius is divided by 3.
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eSolutions Manual - Powered by Cognero
Page 22
2
3
12-6The
Surface
Area
Volumesby
of4Spheres
surface
areaand
is multiplied
or 16. The volume is multiplied by 4 or 64.
The radius is divided by 3.
62/87,21
· 2
3
The surface area is divided by 3 or 9. The volume is divided by 3 or 27.
NAVIGATIONAL COORDINATES Latitude lines circle Earth in an east-west direction.
a. Describe the longest latitude line. What is the geometric name for the circle created by this line?
b. The circumference of Earth is about 25,000 miles. Describe how you can use this to find the surface area of
Earth. Then find the surface area.
c. If two planes are equidistant from the center of a sphere and intersect the sphere, then the circles of intersection
are congruent. Describe how this property can be applied to latitude lines.
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a. The longest latitude line is the line that goes around the middle, called the equator. This line represents a circle
which contains the center of the sphere, so it is a great circle.
b. Find the radius. Then use the formula for surface area of a sphere to find the surface area.
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Page 23
12-6Surface Area and Volumes of Spheres
2
3
The surface area is divided by 3 or 9. The volume is divided by 3 or 27.
NAVIGATIONAL COORDINATES Latitude lines circle Earth in an east-west direction.
a. Describe the longest latitude line. What is the geometric name for the circle created by this line?
b. The circumference of Earth is about 25,000 miles. Describe how you can use this to find the surface area of
Earth. Then find the surface area.
c. If two planes are equidistant from the center of a sphere and intersect the sphere, then the circles of intersection
are congruent. Describe how this property can be applied to latitude lines.
62/87,21
a. The longest latitude line is the line that goes around the middle, called the equator. This line represents a circle
which contains the center of the sphere, so it is a great circle.
b. Find the radius. Then use the formula for surface area of a sphere to find the surface area.
c. Sample answer: Latitude lines the same distance from the center of Earth are congruent circles. So, for example,
WKHODWLWXGHOLQHVUHSUHVHQWLQJƒ1DQGƒ6DUHFRQJUXHQWFLUFOHV7KLVLVWUXHEHFDXVHWKH\DUHHTXLGLVWDQWIURP
WKHHTXDWRU
CHALLENGE A cube has a volume of 216 cubic inches. Find the volume of a sphere that is circumscribed about
the cube. Round to the nearest tenth.
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Since the volume of the cube is 216 cubic inches, the length of each side is 6 inches and so, the length of a diagonal
on a face is
LQFKHV
Use the length of the one side (BC) and this diagonal (AB ) to find the length of the diagonal joining two opposite
vertices of the cube (AC ). The triangle that is formed by these sides is a right triangle. Apply the Pythagorean
Theorem.
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Page 24
c. Sample answer: Latitude lines the same distance from the center of Earth are congruent circles. So, for example,
12-6WKHODWLWXGHOLQHVUHSUHVHQWLQJƒ1DQGƒ6DUHFRQJUXHQWFLUFOHV7KLVLVWUXHEHFDXVHWKH\DUHHTXLGLVWDQWIURP
Surface Area and Volumes of Spheres
WKHHTXDWRU
CHALLENGE A cube has a volume of 216 cubic inches. Find the volume of a sphere that is circumscribed about
the cube. Round to the nearest tenth.
62/87,21
Since the volume of the cube is 216 cubic inches, the length of each side is 6 inches and so, the length of a diagonal
on a face is
LQFKHV
Use the length of the one side (BC) and this diagonal (AB ) to find the length of the diagonal joining two opposite
vertices of the cube (AC ). The triangle that is formed by these sides is a right triangle. Apply the Pythagorean
Theorem.
The sphere is circumscribed about the cube, so the vertices of the cube all lie on the sphere. Therefore, line AC is
also the diameter of the sphere. The radius is
inches. Find the volume.
REASONING Determine whether the following statement is true or false. If true, explain your reasoning. If false,
SURYLGHDFRXQWHUH[DPSOH
If a sphere has radius r, there exists a cone with radius r having the same volume.
62/87,21
Determine if there is a value of h for which the volume of the cone will equal the volume of the sphere.
eSolutions Manual - Powered by Cognero
Page 25
12-6 Surface Area and Volumes of Spheres
REASONING Determine whether the following statement is true or false. If true, explain your reasoning. If false,
SURYLGHDFRXQWHUH[DPSOH
If a sphere has radius r, there exists a cone with radius r having the same volume.
62/87,21
Determine if there is a value of h for which the volume of the cone will equal the volume of the sphere.
The volume of a sphere with radius rLVJLYHQE\
of h LVJLYHQE\
. The volume of a cone with a radius of r and a height
.
Thus, a cone and sphere with the same radius will have the same volume whenever the cone has a height equal to
4r.
Therefore, the statement is true.
OPEN ENDED Sketch a sphere showing two examples of great circles. Sketch another sphere showing two
examples of circles formed by planes intersecting the sphere that are not great circles.
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The great circles need to contain the center of the sphere. The planes that are not great circles need to not contain
the center of the sphere.
Sample answer:
WRITING IN MATH Write a ratio comparing the volume of a sphere with radius r to the volume of a cylinder with
Page 26
what the ratio means.
eSolutions
Manual
by Cognero
radius
r and- Powered
height 2r.
Then describe
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12-6 Surface Area and Volumes of Spheres
WRITING IN MATH Write a ratio comparing the volume of a sphere with radius r to the volume of a cylinder with
radius r and height 2r. Then describe what the ratio means.
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The ratio is 2:3.
The volume of the sphere is two thirds the volume of the cylinder.
GRIDDED RESPONSE What is the volume of the hemisphere shown below in cubic meters?
62/87,21
The volume V of a hemisphere is
RU
, where r is the radius.
Use the formula.
ALGEBRA What is the solution set of 3z + 4 < 6 + 7z?
62/87,21
The correct choice is A.
2
If the area of the great circle of a sphere is 33 ft , what is the surface area of the sphere?
2
F 42 f t
2
G 117
f t - Powered by Cognero
eSolutions
Manual
2
H 132 f t
2
J 264 f t
Page 27
12-6 Surface Area and Volumes of Spheres
The correct choice is A.
2
If the area of the great circle of a sphere is 33 ft , what is the surface area of the sphere?
2
F 42 f t
2
G 117 f t
2
H 132 f t
2
J 264 f t
62/87,21
We know that the area of a great circle is
The surface area S of a sphere is
r is the radius.
Use the formula.
.
, where
So, the correct choice is H.
SAT/ACT If a line LVSHUSHQGLFXODUWRDVHJPHQWAB at E, how many points on line
point A as from point B?
A none
B one
C two
D three
E all points
62/87,21
Since the segment AB is perpendicular to line
correct choice is E.
, all points on
DUHWKHVDPHGLVWDQFHIURP
DUHWKHVDPHGLVWDQFHIURPSRLQWA as from B. The
Find the volume of each pyramid. Round to the nearest tenth if necessary.
62/87,21
7KHEDVHRIWKHS\UDPLGLVDVTXDUHZLWKDVLGHRIIHHW7KHVODQWKHLJKWRIWKHS\UDPLGLVIHHW8VHWKH
Pythagorean Theorem to find the height h.
eSolutions Manual - Powered by Cognero
Page 28
E all points
62/87,21
the segment
is perpendicular
to line
12-6Since
Surface
Area andABVolumes
of Spheres
correct choice is E.
, all points on
DUHWKHVDPHGLVWDQFHIURPSRLQWA as from B. The
Find the volume of each pyramid. Round to the nearest tenth if necessary.
62/87,21
7KHEDVHRIWKHS\UDPLGLVDVTXDUHZLWKDVLGHRIIHHW7KHVODQWKHLJKWRIWKHS\UDPLGLVIHHW8VHWKH
Pythagorean Theorem to find the height h.
.
The volume of a pyramid is
, where B is the area of the base and h is the height of the pyramid.
3
Therefore, the volume of the pyramid is about 58.9 ft .
62/87,21
The volume of a pyramid is
, where B is the area of the base and h is the height of the pyramid. The base
of this pyramid is a right triangle with a leg of 8 inches and a hypotenuse of 17 inches. Use the Pythagorean
Theorem to find the length of the other leg a .
eSolutions Manual - Powered by Cognero
Page 29
3
12-6 Surface Area and Volumes of Spheres
Therefore, the volume of the pyramid is about 58.9 ft .
62/87,21
The volume of a pyramid is
, where B is the area of the base and h is the height of the pyramid. The base
of this pyramid is a right triangle with a leg of 8 inches and a hypotenuse of 17 inches. Use the Pythagorean
Theorem to find the length of the other leg a .
The height of the pyramid is 12 inches.
3
Therefore, the volume of the pyramid is 240 in .
62/87,21
The base of this pyramid is a rectangle of length 10 meters and width 6 meters. The slant height of the pyramid is 12
meters. Use the Pythagorean Theorem to find the height h .
eSolutions Manual - Powered by Cognero
Page 30
12-6Surface Area and Volumes of Spheres
3
Therefore, the volume of the pyramid is 240 in .
62/87,21
The base of this pyramid is a rectangle of length 10 meters and width 6 meters. The slant height of the pyramid is 12
meters. Use the Pythagorean Theorem to find the height h .
The volume of a pyramid is
, where B is the area of the base and h is the height of the pyramid.
Therefore, the volume of the pyramid is about 232.4 m3.
ENGINEERING The base of an oil drilling platform is made up of 24 concrete cylindrical cells. Twenty of the cells
are used for oil storage. The pillars that support the platform deck rest on the four other cells. Find the total volume
of the storage cells.
62/87,21
eSolutions Manual - Powered by Cognero
The volume of a cylinder is
, where r is the radius and h is the height of the cylinder.
Page 31
12-6Surface Area and Volumes of Spheres
Therefore, the volume of the pyramid is about 232.4 m3.
ENGINEERING The base of an oil drilling platform is made up of 24 concrete cylindrical cells. Twenty of the cells
are used for oil storage. The pillars that support the platform deck rest on the four other cells. Find the total volume
of the storage cells.
62/87,21
The volume of a cylinder is
, where r is the radius and h is the height of the cylinder.
3
The total volume of the storage cells is about 18,555,031.6 ft .
Find the area of each shaded region. Round to the nearest tenth.
62/87,21
:HDUHJLYHQWKHEDVHRIWKHWULDQJOH6LQFHWKHWULDQJOHLVƒ-ƒ-ƒWKHKHLJKWLVDOVR
7KHWULDQJOHLVDƒ-ƒ-ƒWULDQJOHVRWKHGLDPHWHURIWKHFLUFOHLV
and the radius h is
.
eSolutions Manual - Powered by Cognero
Page 32
12-6 Surface Area and Volumes of Spheres
3
The total volume of the storage cells is about 18,555,031.6 ft .
Find the area of each shaded region. Round to the nearest tenth.
62/87,21
:HDUHJLYHQWKHEDVHRIWKHWULDQJOH6LQFHWKHWULDQJOHLVƒ-ƒ-ƒWKHKHLJKWLVDOVR
7KHWULDQJOHLVDƒ-ƒ-ƒWULDQJOHVRWKHGLDPHWHURIWKHFLUFOHLV
and the radius h is
.
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The side length of the given square is 2(7) or 14 units.
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Page 33
12-6 Surface Area and Volumes of Spheres
62/87,21
The side length of the given square is 2(7) or 14 units.
62/87,21
$UHJXODURFWDJRQKDVFRQJUXHQWFHQWUDODQJOHVVRWKHPHDVXUHRIHDFKFHQWUDODQJOHLV· Apothem
LVWKHKHLJKWRIWKHLVRVFHOHVWULDQJOH$%&8VHWKH7ULJRQRPHWULFUDWLRVWRILQGWKHVLGHOHQJWKDQG
apothem of the polygon.
eSolutions Manual - Powered by Cognero
Page 34
12-6 Surface Area and Volumes of Spheres
AB = 2(DB) = 32 sin 22.5
COORDINATE GEOMETRY Find the area of each figure.
WXYZ with W(0, 0), X (4, 0), Y(5, 5), and Z(1, 5)
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Graph the diagram.
7KHOHQJWKRIWKHEDVHRIWKHSDUDOOHORJUDPJRHVIURP WR VRLWLVXQLWV
7KHKHLJKWRIWKHSDUDOOHORJUDPJRHVIURP WR VRLWLVXQLWV
The area of a parallelogram is the product of a base b and its corresponding height h . Therefore, the area is 20
2
units .
ABC with A (2, ±3), B(±5, ±3), and C(±1, 3)
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62/87,21
Graph the diagram.
Page 35
7KHKHLJKWRIWKHSDUDOOHORJUDPJRHVIURP WR VRLWLVXQLWV
The area of a parallelogram is the product of a base b and its corresponding height h . Therefore, the area is 20
2
12-6units
Surface
. Area and Volumes of Spheres
ABC with A (2, ±3), B(±5, ±3), and C(±1, 3)
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Graph the diagram.
The length of the base of the triangle goes from (±5, ±3) to (2, ±3), so it is 7 units. The height of the triangle goes
from (±1, ±3) to (±1, 3), so it is 6 units.
2
Therefore, the area is 0.5(7)(6) = 21 units .
Refer to the figure.
How many planes appear in this figure?
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A plane is a flat surface made up of points that extends infinitely in all directions. There is exactly one plane
through any three points not on the same line.
plane P
plane ABD
plane ACD
plane ABC
Name three points that are collinear.
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Collinear points are points that lie on the same line. Noncollinear points do not lie on the same line.
D, B, and G lie on line DG.
Are points G, A , B, and E coplanar? Explain.
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Coplanar points are points that lie in the same plane. Noncoplanar points do not lie in the same plane.
Points A , B, and E lie in plane P, but point G does not lie in plane P. Thus, they are not coplanar. Points A , G, and B
lie in plane AGB , but point E does not lie in plane AGB .
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At what point do
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and
LQWHUVHFW"
Page 36
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Collinear points are points that lie on the same line. Noncollinear points do not lie on the same line.
12-6Surface Area and Volumes of Spheres
D, B, and G lie on line DG.
Are points G, A , B, and E coplanar? Explain.
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Coplanar points are points that lie in the same plane. Noncoplanar points do not lie in the same plane.
Points A , B, and E lie in plane P, but point G does not lie in plane P. Thus, they are not coplanar. Points A , G, and B
lie in plane AGB , but point E does not lie in plane AGB .
At what point do
and
LQWHUVHFW"
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and
GRQRWLQWHUVHFW
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lies in plane P, but only E lies in P. If E were on
, then they would intersect.
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