SURFACE AREA AND VOLUME OF SPHERES

About "Surface Area and Volume of Spheres"

Surface Area and Volume of Spheres :

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

Finding the Surface Area of a Sphere

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  =  4πr2

Finding the Volume of a Sphere

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 4πr2. 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 4πr2 for nB.

V  ≈  1/3 ⋅ (4πr2)r

Simplify.

V  =  4/3 ⋅ πr3

Theorem (Volume of a Sphere) :

The volume V of a sphere with radius r is

V  =  4/3 ⋅ πr3

Great Circle of a Sphere

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. 

Finding the Surface Area of a Sphere

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  =  4πr2

Substitute 2 for r. 

S  =  4π (22)

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  =  4πr2

Substitute 4 for r. 

S  =  4π (42)

S  =  4π (16)

S  =  64π  in2

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.

Using a Great Circle 

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  =  4πr2

Substitute 6.9 for r. 

S  =  4π(6.9)2

Simplify.

S  =  4π( 47.61)

Use calculator. 

S  ≈  598  ft2

So, the surface area of the sphere is about 598 square feet.

Finding the Volume of a Sphere 

Example : 

Find the volume of the sphere shown below. 

Solution : 

Formula for volume of a sphere : 

V  =  4/3 ⋅ πr3

Substitute 22 for r. 

V  =  4/3 ⋅ π(223)

Simplify.

V  =  4/3 ⋅ π(10648)

V  =  42592/3 ⋅ π

Use calculator.

V    44602 cm2

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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