11.6 Notes.notebook
April 18, 2013
The three orange slices below were cut from three different oranges. Do you have sufficient information to tell which orange is the largest? If not, explain what information you would need.
No; you need to know how far from the center of the orange each slice was cut.
Apr 16­7:01 PM
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11.6 Notes.notebook
April 18, 2013
11.6 Surface Areas and Volumes of Spheres
A sphere with center O and radius r is the set of all points in space at a distance r from point O.
O
r
Apr 6­5:17 PM
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11.6 Notes.notebook
April 18, 2013
A sphere is the set of all points in space equidistant from a given point called the center.
A radius is a segment that has one endpoint at the center and the other endpoint on the sphere.
O
r
A diameter is a segment passing through the center with endpoints on the sphere.
Apr 16­7:04 PM
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11.6 Notes.notebook
April 18, 2013
You can find the surface area and the volume of a sphere when you know its radius.
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11.6 Notes.notebook
April 18, 2013
When a plane and a sphere intersect in more than one point, the intersection is a circle.
If the center of the circle is also the center of the sphere, it is called a great circle.
Apr 16­7:05 PM
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11.6 Notes.notebook
April 18, 2013
The circumference of a great circle is the circumference of the sphere.
A great circle divides a sphere into two hemispheres.
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11.6 Notes.notebook
April 18, 2013
A baseball can model a sphere. To approximate its surface area, you can take apart its covering. Each of the two pieces suggests a pair of circles with radius r, which is approximately the radius of the ball. The area of the four circles, 4πr2, suggest the surface area of the ball.
Apr 16­7:06 PM
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11.6 Notes.notebook
April 18, 2013
Surface Area of a Sphere
The surface area of a sphere is four times the product of π and the square of the radius of the sphere.
S.A. = 4πr2
O
r
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11.6 Notes.notebook
April 18, 2013
What is the surface area of the sphere in terms of π?
100 π m2
Apr 16­7:07 PM
9
11.6 Notes.notebook
April 18, 2013
What is the surface area of a sphere with a diameter of 14 in? Give your answer in terms of π and rounded to the nearest square inch.
196 π in2; 616 in2
Apr 16­7:07 PM
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11.6 Notes.notebook
April 18, 2013
What is the surface area of the sphere in terms of π?
18 in.
324 π in2
Apr 16­7:16 PM
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11.6 Notes.notebook
April 18, 2013
You can use spheres to approximate the surface areas of real­world objects.
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11.6 Notes.notebook
April 18, 2013
Earth's equator is about 24,902 mi long. What is the approximate surface area of Earth? Round to the nearest thousand square miles.
197,387,000 mi2
Apr 16­7:07 PM
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11.6 Notes.notebook
April 18, 2013
What is the surface area of a melon with circumference 18 in? Round your answer to the nearest ten square inches.
100 in2
Apr 16­7:08 PM
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11.6 Notes.notebook
April 18, 2013
What is the surface area of a ball with a circumference of about 25 in? Round to the nearest ten square inches.
200 in2
Apr 16­7:18 PM
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11.6 Notes.notebook
April 18, 2013
Recall that the volume of a cone is 1/3πr3.
You can use this with Cavalieri's Principle to find the formula for the volume of a sphere.
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11.6 Notes.notebook
April 18, 2013
Both figures have a parallel plane x units above their centers that form circular cross sections.
Apr 16­7:08 PM
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11.6 Notes.notebook
April 18, 2013
The area of the cross section of the cylinder minus the area of the cross section of the cone is the same as the area of the cross section of the sphere.
Every horizontal plane will cut the figures into cross sections of equal area. Apr 16­7:08 PM
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11.6 Notes.notebook
April 18, 2013
By Cavalieri's Principle, the volume of the sphere = the volume of the cylinder ­
the volume of 2 cones
Volume of a sphere = πr2(2r) ­ 2(1/3πr3)
= 2πr3 ­ 2/3πr3 = 4/3πr3
Apr 16­7:08 PM
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11.6 Notes.notebook
April 18, 2013
Volume of a Sphere
The volume of a sphere is four thirds the product of π and the cube of the radius of the sphere.
V = 4/3πr2
O
r
Apr 16­7:09 PM
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11.6 Notes.notebook
April 18, 2013
What is the volume of the sphere in terms of π?
288π m3
Apr 16­7:10 PM
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11.6 Notes.notebook
April 18, 2013
A sphere has diameter of 60 in. What is its volume to the nearest cubic inch.
113,097 in3
Apr 16­7:10 PM
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11.6 Notes.notebook
April 18, 2013
Suppose the radius of a sphere is halved. How does this affect the volume of the sphere. Explain.
Apr 16­7:11 PM
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11.6 Notes.notebook
April 18, 2013
What is the volume of the sphere in terms of π?
12 mm
2304π mm3
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11.6 Notes.notebook
April 18, 2013
Find the area and the volume of a sphere with radius 2 cm.
O
r
A = 16π cm2
V = 32π/3 cm3
Apr 6­5:17 PM
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11.6 Notes.notebook
April 18, 2013
Notice that you only need to know the radius of a sphere to find its volume and surface area. This means that if you know the volume of a sphere, you can find its surface area.
Apr 16­7:11 PM
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11.6 Notes.notebook
April 18, 2013
The volume of a sphere is 5000 m3. What is its surface area to the nearest square meter?
1414 m2
Apr 16­7:11 PM
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11.6 Notes.notebook
April 18, 2013
The volume of a sphere is 4200 ft3. What is its surface area to the nearest tenth?
1258.9 ft2
Apr 16­7:11 PM
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11.6 Notes.notebook
April 18, 2013
The volume of a sphere is 2200 yd3. What is the surface area of the sphere?
818 yd2
Apr 16­7:18 PM
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11.6 Notes.notebook
April 18, 2013
The diameter of a sphere is 12 ft.
What is its surface area in terms of π? What is its volume to the nearest tenth?
144π ft2
904.8 ft3
Apr 16­7:12 PM
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11.6 Notes.notebook
April 18, 2013
The volume of a sphere is 80π cm3. What is its surface area to the nearest whole number?
193 cm2
Apr 16­7:12 PM
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11.6 Notes.notebook
April 18, 2013
What is the ratio of the area of a great circle to the surface area of the sphere?
1:4
Apr 16­7:12 PM
32
11.6 Notes.notebook
April 18, 2013
The area of a sphere is 256π. Find the volume.
O
V = 2048
r
π/3
Apr 6­5:17 PM
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11.6 Notes.notebook
April 18, 2013
A plane passes 4 cm from the center of a sphere with radius 7 cm. Find the area of the circle of intersection.
A = 33π cm2
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