A circle was described as the locus of points in a plane that are a given distance from a point. A sphere is the locus of points in space that are a given distance from a point. The point is called the center of the sphere. 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. A diameter is a chord that contains the center. As with circles, the terms radius and diameter also represent distances, and the diameter is twice the radius.
Theorem (Surface Area of a Sphere) :
The surface area S of a sphere with radius r is
S = 4πr^{2}
Imagine that the interior of a sphere with radius r is approximated by n pyramids, each with a base area of B and a height of r, as shown below.
The volume of each pyramid is
= 1/3 ⋅ Br
and the sum of the base areas is
= nB
The surface area of the sphere is approximately equal to nB, or 4πr^{2}. So, we can approximate the volume V of the sphere as follows.
Each pyramid has a volume of 1/3 ⋅ Br
V ≈ n ⋅ 1/3 ⋅ Br
Regroup factors.
V = 1/3 ⋅ (nB)r
Substitute 4πr^{2 }for nB.
V ≈ 1/3 ⋅ (4πr^{2})r
Simplify.
V = 4/3 ⋅ πr^{3}
Theorem (Volume of a Sphere) :
The volume V of a sphere with radius r is
V = 4/3 ⋅ πr^{3}
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.
Example 1 :
(a) Find the surface area of the sphere shown below.
(b) When the radius doubles, does the surface area double ?
Solution :
Solution (a) :
Formula for surface area of a sphere :
S = 4πr^{2}
Substitute 2 for r.
S = 4π (2^{2})
S = 4π (4)
S = 16π
The surface area of the sphere is 16π square inches.
Solution (b) :
When the radius doubles,
r = 2 ⋅ 2
r = 4 inches
Formula for surface area of a sphere :
S = 4πr^{2}
Substitute 4 for r.
S = 4π (4^{2})
S = 4π (16)
S = 64π in^{2}
Because 16π ⋅ 4 = 64π, the surface area of the sphere in part (b) is four times the surface area of the sphere in part (a).
So, when the radius of a sphere doubles, the surface area does not double.
Example 2 :
The circumference of a great circle of a sphere is 13.8π feet. What is the surface area of the sphere ?
Solution :
Draw a sketch.
Begin by finding the radius of the sphere.
Formula for circumference of a circle :
C = 2πr
Substitute 13.8π for C.
13.8π = 2πr
Divide each side by 2π.
6.9 = r
Formula for surface area of a sphere :
S = 4πr^{2}
Substitute 6.9 for r.
S = 4π(6.9)^{2}
Simplify.
S = 4π( 47.61)
Use calculator.
S ≈ 598 ft^{2}
So, the surface area of the sphere is about 598 square feet.
Example 3 :
Find the volume of the sphere shown below.
Solution :
Formula for volume of a sphere :
V = 4/3 ⋅ πr^{3}
Substitute 22 for r.
V = 4/3 ⋅ π(22^{3})
Simplify.
V = 4/3 ⋅ π(10648)
V = 42592/3 ⋅ π
Use calculator.
V ≈ 44602 cm^{2}
The volume of the sphere is about 44602 cubic cm.
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