[Maths Class Notes] on Sphere Formula Pdf for Exam

What is the Sphere?

A sphere can be regarded as an absolute symmetrical circular shaped object in a three dimensional space. In a three dimensional space, all the points on the surface of the sphere are at the same distance from a fixed point which is regarded as the center of the sphere. The straight line that connects the center of the sphere to any point on its surface is called the radius of the sphere which is generally represented by the letter ‘r’. Diameter of a sphere is that longest line which passes through the center of the sphere and touches its surface at two different points. 

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Important Sphere formula

1. Diameter of a Sphere:

The diameter of a sphere is the straight line that is passing through the center of the sphere and touches two points on either side of its surface. The diameter of a sphere is always two times its radius. If the radius of the sphere is ‘r’, then its diameter is given by the formula:

D = 2 x r

2. Circumference of the Sphere:

Circumference of a sphere can be calculated as 2π times its radius. Circumference of a sphere and that of a circle is given by the same formula:

C = 2 π r

Here, π is a constant and its value is 3.14 or 22/7. So, the circumference of a sphere can also be computed as 6.28 times or 44/7 times its radius. 

3. Total Surface Area of Sphere

The total surface area of a sphere is the same as its curved surface area because the sphere does not have any lateral surfaces. The formula obtained by deriving surface area of a sphere is written Mathematically as:

TSA = 4 π r2

In the above equation, 

TSA is the total surface area of a sphere. It can be simply stated as surface area. 

π is a constant and its value is equal to 3.14 or 22/7

‘r’ represents the value of the radius of the given sphere 

So, the formula of deriving surface area of a sphere is equal to 4π times or 12.56 times or 88/7 times the square of the radius of the sphere. 

4. Volume of a Sphere

Deriving volume of a sphere is the same as finding the total space available within the surface of the sphere. The mathematical formula of deriving volume of a sphere is given as:

V = 4/3 π r3

In the above equation,

‘V’ is the volume of the sphere

π is a constant and its value is equal to 3.14 or 22/7

‘r’ represents the value of the radius of the given sphere 

So, the formula for the deriving volume of a sphere can be stated as 4π/3 times or 4.19 times or 88/21 times the cube of the radius of the sphere whose volume is to be determined.

Worked Examples of Sphere Formula

1. Calculate the diameter and the circumference of a sphere whose radius is 7 cm.

Solution:

Given: Radius of the sphere = 7cm

Diameter of the sphere is calculated as:

D = 2 x r

D = 2 x 7 

D = 14 cm

Circumference of the sphere is found by the formula

C = 2 x π x r

C = 2 x (22/7) x 7

C = 2 x 22 

C = 44 cm

Therefore, the diameter and circumference of the sphere are 14 cm and 44 cm respectively.

2. Find the total surface area and the volume of a sphere whose radius is 14 cm.

Solution:

Given: Radius of the sphere = 14 cm

The formula for the deriving surface area of a sphere is:

A = 4 π r2

A = 4  x (22/7) x (14)2

A = 4 x (22/7) x 14 x 14

A = 4 x 22 x 28

A = 2464 cm2

The volume of a sphere is found using the formula:

V = (4/3) π r3

V = (4/3) x (22/7) x (14)3

V = 11494.04 cc

Therefore, the volume and total surface area of a sphere of radius 14 cm are 11494.04 cc and 2464 cm2 respectively.

3. The volume of a sphere is found to be 729 cc. Find its radius.

Solution:

Given: Volume of the sphere = 729 cc

The formula for deriving volume of a sphere is 

V = 4/3 π r3

729 = (4/3) (22/7) r3

729 = (88/21) r3

r3 = (729 x 21) / 88

r3 = 173.97

r = ∛173.97

r = 5.58 cm

Therefore, the radius of the sphere is 5.58 cm

Fun Facts About What is the Sphere

• By deriving surface area of a sphere formula, it was found by Archimedes that it was the same as the lateral surface area of a cylinder with the base radius equal to that of the sphere and the height equal to the diameter of the sphere. 

• The sphere and circle are not the same. The circle is a two-dimensional closed plane geometric figure whereas a sphere is a three-dimensional circle.