Spheres

1
Lesson 7.6

• Spheres

Bellringer Using your calculator, find the cube
root of 33.
2
Spheres
Definition
In space, the set of all points that are a given
distance from a given point, called the center.
A sphere is formed by revolving a circle about
its diameter.
3
Spheres special segments lines
Radius A segment whose endpoints are the center
of the sphere and a point on the sphere.
Chord A segment whose endpoints are on the
sphere.
Diameter A chord that contains the spheres
center.
Tangent A line that intersects the sphere in
exactly one point.
Diameter
Tangent
Chord
Radius
4
Great Circle Hemisphere
Great Circle For a given sphere, the
intersection of the sphere and a plane that
contains the center of the sphere.
Hemisphere One of the two parts into which a
great circle separates a given sphere.
Great Circle
Hemisphere
5
Lesson Videos
Volume
Surface Area
6
Ex. 1 Finding the Surface Area of a Sphere

• Find the surface area. When the radius doubles,
  does the surface area double?

7

• S 4?r2
• 4?22
• 16? in.2

• S 4?r2
• 4?42
• 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.
8
Ex. 2 Using a Great Circle

• The circumference of a great circle of a sphere
  is 13.8? feet. What is the surface area of the
  sphere?

9
Solution

• Begin by finding the radius of the sphere.
• C 2?r
• 13.8? 2?r
• 13.8?
• 2?r
• 6.9 r

r
10
Solution

• Using a radius of 6.9 feet, the surface area is
  
• S 4?r2
• 4?(6.9)2
• 190.44? ft.2

So, the surface area of the sphere is 190.44 ?
ft.2
11
Ex. 3 Finding the Surface Area of a Sphere

• Baseball. A baseball and its leather covering
  are shown. The baseball has a radius of about
  1.45 inches.
• Estimate the amount of leather used to cover the
  baseball.
• The surface area of a baseball is sewn from two
  congruent shapes, each which resembles two joined
  circles. How does this relate to the formula for
  the surface area of a sphere?

12
Ex. 3 Finding the Surface Area of a Sphere
13
Theorem 7.6.1 Volume of a Sphere

• The volume of a sphere with radius r is S

3
14
Ex. 4 Finding the Volume of a Sphere

• Ball Bearings. To make a steel ball bearing, a
  cylindrical slug is heated and pressed into a
  spherical shape with the same volume. Find the
  radius of the ball bearing to the right

15
Solution

• To find the volume of the slug, use the formula
  for the volume of a cylinder.
• V ?r2h
• ?(12)(2)
• 2? cm3
• To find the radius of the ball bearing, use the
  formula for the volume of a sphere and solve for
  r.

16
More . . .

• V 4/3?r3
• 2? 4/3?r3
• 6? 4?r3
• 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.

So, the radius of the ball bearing is about 1.14
cm.
17
Homework

• Page 474/ 9, 15, 27, 31, 36, 37