To find the surface area and volume - nhsrobinson.weebly.com · standard ball? standard ball: ......
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Transcript of To find the surface area and volume - nhsrobinson.weebly.com · standard ball? standard ball: ......
To find the surface
area and volume
of a sphere
Sphere – set of all points in space
equidistant from a given point called the
center.
r
Surface Area Formula:
S.A. = 4πr2
Volume Formula:V = πr3
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Great circle – the intersection of a
sphere and a plane containing the
center of the sphere
r
The great circle cuts the
sphere into two hemispheres.
The circumference of a great
circle is the circumference of the
sphere.
Holt McDougal Geometry
11-4 Spheres
Example 1A: Finding Volumes of Spheres
Find the volume of the sphere. Give your answer in terms of .
= 2304 in3 Simplify.
Volume of a sphere.
Substitute 24
2= 12 for r.
Holt McDougal Geometry
11-4 Spheres
Example 1B: Finding Volumes of Spheres
Find the diameter of a sphere with volume 36,000 cm3.
Substitute 36,000 for V.
27,000 = r3
r = 30
d = 60 cm d = 2r
Take the cube root of both sides.
Volume of a sphere.
Holt McDougal Geometry
11-4 Spheres
Example 1C: Finding Volumes of Spheres
Find the volume of the hemisphere.
Volume of a hemisphere
Substitute 15 for r.
= 2250 m3 Simplify.
Holt McDougal Geometry
11-4 Spheres
Example 2: Sports Application
A sporting goods store sells exercise balls in two sizes, standard (22-in. diameter) and jumbo (34-in. diameter). How many times as great is the volume of a jumbo ball as the volume of a standard ball?
standard ball:jumbo ball:
A jumbo ball is about 3.7 times as great in volume as a standard ball.
Holt McDougal Geometry
11-4 Spheres
Example 3A: Finding Surface Area of Spheres
Find the surface area of a sphere with diameter 76 cm. Give your answers in terms of .
S = 4r2
S = 4(38)2 = 5776 cm2
Surface area of a sphere
Holt McDougal Geometry
11-4 Spheres
Example 3B: Finding Surface Area of Spheres
Find the volume of a sphere with surface area 324 in2. Give your answers in terms of .
Substitute 324 for S.324 = 4r2
r = 9 Solve for r.
Substitute 9 for r.
The volume of the sphere is 972 in2.
S = 4r2 Surface area of a sphere
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Holt McDougal Geometry
11-4 Spheres
Example 3C: Finding Surface Area of Spheres
Find the surface area of a sphere with a great circle that has an area of 49 mi2.
Substitute 49 for A.49 = r2
r = 7 Solve for r.
S = 4r2
= 4(7)2 = 196 mi2 Substitute 7 for r.
A = r2 Area of a circle
Holt McDougal Geometry
11-4 Spheres
Example 4: Exploring Effects of Changing Dimensions
The radius of the sphere is
multiplied by . Describe the
effect on the volume.
original dimensions: radius multiplied by :
Notice that . If the radius is multiplied
by , the volume is multiplied by , or .
Holt McDougal Geometry
11-4 Spheres
Example 5: Finding Surface Areas and Volumes of
Composite Figures
Find the surface area and volume of the composite figure. Give your answer in terms of .
Step 1 Find the surface area of the composite figure.
The surface area of the composite figure is the sum of the curved surface area of the hemisphere, the lateral area of the cylinder, and the base area of the cylinder.
Holt McDougal Geometry
11-4 Spheres
Example 5 Continued
The surface area of the composite figure is
L(cylinder) = 2rh = 2(6)(9) = 108 in2
B(cylinder) = r2 = (6)2 = 36 in2
72 + 108 + 36 = 216 in2.
Find the surface area and volume of the composite figure. Give your answer in terms of .
Holt McDougal Geometry
11-4 Spheres
Step 2 Find the volume of the composite figure.
Example 5 Continued
Find the surface area and volume of the composite figure. Give your answer in terms of .
The volume of the composite figure is the sum of the volume of the hemisphere and the volume of the cylinder.
The volume of the composite figure is 144 + 324 = 468 in3.
Holt McDougal Geometry
11-4 Spheres
Lesson Quiz: Part I
Find each measurement. Give your answers in terms of .
1. the volume and surface area
of the sphere
2. the volume and surface area of a sphere with
great circle area 36 in2
3. the volume and surface area
of the hemisphere
V = 36 cm3; S = 36 cm2
V = 288 in3;S = 144 in2
V = 23,958 ft3;S = 3267 ft2
Holt McDougal Geometry
11-4 Spheres
Lesson Quiz: Part II
4. A sphere has radius 4. If the radius is multiplied
by 5, describe what happens to the surface area.
5. Find the volume and surface area of the
composite figure. Give your answer in terms of .
The surface area is multiplied by 25.
V = 522 ft3; S = 267 ft2
Formulas for
spheres:
S.A. = 4πr2
V = πr34
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