226 Part 3: The Shape of the World
A sphere is the set of all points in
space at a fixed distance from a
center point.Surface Area
The surface area of a sphere depends on the radius of the
sphere. The formula SA = 4Sr^2 gives you the surface area of
a sphere of radius r.
An official soccer ball is a sphere with a circumference of 68
to 70 centimeters. That means its diameter is^68 to^70 centi-
meters, so its radius is^34 to^35
centimeters. That’s between
10.8 and 11.1 centimeters. The surface area of the official
ball will be at least 4 34 4 34 4 34 1471.9(^22)
2
2
^
square centimeters and not more than
4 35 4 35 4 35 1559.7
(^22)
2
2
square centimeters.
Volume
The volume of a sphere, like the surface area, depends on the radius of the sphere. The formula
for the volume of a sphere is Vr^4
3
^3. Volume is measured in cubic units, so it’s not surprising
that the radius is cubed. And given how round it is, the presence of S isn’t a surprise, either.
Unfortunately, explaining where the
4
3
came from requires more advanced math than I can
include here, so you’ll have to just believe me for that part.
The radius of the earth is about 3,959 miles. To find the volume of your home planet, calculate
Vr^4
3
4
3
(^3) (^39593).
82 billion times pi is a very large number. It’s approximately 2.6 v 1011 or about 260 billion cubic
miles.
Vr^4
3
4
3
3959 4
3
(^3) ^3 62,052,103,080 82,736,137,440
CHECK POINT
- Find the surface area of a sphere with a radius of 8 inches.
- Find the volume of a sphere with a radius of 12 cm.
- Find the surface area of a sphere with a diameter of 4 m.
- Find the volume of a sphere with a diameter of 6 feet.
- Find the radius of a sphere with a volume of 4500S cubic centimeters.