surface area and volume of spheres a sphere is the set of all points in space that are the same...
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Surface Area and Volume of Spheres
• A sphere is the set of all points in space that are the same distance from a point, the center of the sphere.
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• A radius of a sphere is a segment from the center to a point on the sphere.
• A chord of a sphere is a segment whose endpoints are on the sphere.
Surface Area and Volume of Spheres
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• A diameter is a chord that contains the center. As with all circles, the terms radius and diameter also represent distances, and the diameter is twice the radius.
Surface Area and Volume of Spheres
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More . . .
• If a plane intersects a sphere, the intersection is either a single point or a circle. If the plane contains the center of the sphere, then the intersection is a great circle of the sphere. Every great circle of a sphere separates a sphere into two congruent halves called hemispheres.
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Surface Area of a Sphere
• Surface area = 4 π (radius)2
• S = 4 π r2
C Radius
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Find the Surface Area of a Sphere• Find the surface area of the sphere. Round your
answer to the nearest whole number.
The radius is 8 in.
24S r24 (8)
24 64 804 .in
8 in.
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Find the Surface Area of a Sphere
• Find the surface area of the sphere. Round your answer to the nearest whole number.
The radius is 5 cm.
24S r24 (5)
24 25 314 .cm
10 cm.
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Ex. 1: Comparing Surface Areas
• Find the surface area. When the radius doubles, does the surface area double?
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S = 4r2
= 422
= 16 in.2
S = 4r2
= 442
= 64 in.2
The surface area of the sphere in part (b) is four times greater than the surface area of the sphere in part (a) because 16 • 4 = 64
So, when the radius of a sphere doubles, the surface area DOES NOT double.
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Ex. 2: Using a Great Circle
• The circumference of a sphere (also referred to as the circumference of the great circle) is 13.8 feet. What is the surface area of the sphere?
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Solution:
Begin by finding the radius of the sphere.
C = 2r
13.8 = 2r13.8 = r (To solve for r, divide both sides by 2)
2 6.9 = r
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Solution:
Using a radius of 6.9 feet, the surface area is:
S = 4r2
= 4(6.9)2
= 190.44 ft.2
So, the surface area of the sphere is 190.44 feet squared.
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Ex. 3: Finding the Surface Area of a Sphere
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Volume of a Sphere
34
3V r
radius
C
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Find the Volume of a Sphere• Find the volume of the sphere. Round your answer to
the nearest whole number.
a. 34
3V r
34(2)
3
334 .ft
2 ft.
4 Ab/c 3 2 ^ 3 =
Using your calculator, input:
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Find the Volume of a Sphere
Find the volume of the hemisphere.
Round your answer to the nearest whole number.
Note: A hemisphere has HALF
the volume of a sphere.
• So simply use the volume formula,
then ½ the answer.
3262 .in
a5 in.
Using your calculator, input: 4 Ab/c 3 5 ^ 3 =
34
3V r
≈523.95877 / 2 = ≈262
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Find the Volume of a Sphere
• Find the volume of the sphere. Write your answer in terms of .
34
3V r
34(6)
3
2883905 .in
6 in.
Be sure to remember to include when you write your answer
288x
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Last Example: Find r, if V = 2V = 4 r3
3
2 = 4 r3
3
6 = 4r3
1.5 = r3
1.14 r
Formula for volume of a sphere.
Substitute 2 for V.
Multiply each side by 3.
Divide each side by 4.
Use a calculator to take the cube root.
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The End
• Assignment: 6.9 NTG p. 245 – 251 all
• VOCABULARY on p. 245 –Sketch picture or write definition.
• NOTES PAGES p. 246-248 – Read and fill in the blanks.
• PRACTICE p. 249-251 – Complete 1 – 33 all.
• If you get stuck, refer back to your Presentation Notes.