**Surface Area and Volume of Spheres :**

In this section, we are going to see, how to find surface area and volume of spheres.

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.

After having gone through the stuff given above, we hope that the students would have understood, "Surface Area and Volume of Spheres".

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