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 :**

(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 :**

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 : **

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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