Area & Volume Of Sphere

A sphere is a solid figure bounded by a curved surface such that every point on the surface is the same distance from the center.   In other words, a sphere is a perfectly round geometrical object in three-dimensional space, just like a surface of a round ball. The surface area of a sphere is defined as the amount of region covered by the surface of a sphere and is equal to 4πr².

A sphere is mathematically defined as the set of points that are all at the same distance from a given point but in three-dimensional space.

The distance from the center to the edge is called the radius of the ball, and the line that connects two points on the sphere and is twice the length of the radius is called as diameter.

Area and Volume Of Sphere

Area & Volume Of Sphere

Surface Area of Sphere

A sphere is the 3- D shape where the curved surface area equals to the total surface area of the figure. The curved surface area is the area in which only the area of the curved part is covered. The formula does not take into account the circular base

Curved Surface Area = 2πr²

The total surface area, on the other hand, is a combination of the curved area along with the area of the base The formula for the total surface area  and the curved surface area of the sphere is given below

Surface area (TSA) = CSA = 4πr²

How do you Find the Total  Surface Area of a Sphere?

To find the area of the sphere firstly, find the radius of the sphere. Secondly, apply the formula and simplify.

Volume Of Sphere

The volume of the sphere is defined as the number of cubic units needed to fill a sphere.

S.I unit is given cubic meters ( m³)

The derivation is given as:

Volume Of Sphere

v=Volume of the Sphere

 = Sum of the volumes of all pyramids

= 1/3A1r + 1/3A2r  + 1/3A3r+….+1/3Anr

= 1/3(Surface area of the sphere) r

=1/3(4πr2)×r

=4/3(πr3)

In the above derivation, the sphere in Fig:2 is divided into pyramids, and the volume of the sphere is equal to the volume of all the pyramids as shown in the figure. The total volume is calculated by the summation of the pyramids volumes.

Steps for Finding the Volume of a Sphere

Step 1: Find the radius of a sphere and cubic it.

Step 2: Take the product of 4/3 π and the cube of the radius of the sphere.

Step 3: Write answer in the proper cubic unit of measurement.

Examples

Example 1: Find the volume of a sphere of diameter 12 m, rounding your answer to two decimal places(using pi = 3.14).

Solution:

Diameter of the sphere (d) = 12 m

Step 1:

Find the radius of the sphere:

Radius of the sphere (r) = d/2 = 12/2 = 6 m

Step 2:

Volume of the sphere (V) = 4/3 π r3

V = 4/3 x 3.14 x 63

V = 4/3 x 3.14 x 6 x 6 x 6

V = 904.32

Hence the volume of the sphere is 904.32 m3 .

Example 2:

A plane passes through the center of a sphere and forms a circle with a radius of 12 feet. What is the surface area of the sphere.

Solution:

Radius of the sphere = 12 feet

Volume of the sphere (V) = 4 π r2 

= 4 x 3.14 x 12 x 12

= 1808.64

Hence area of the sphere is 1808.64 feet2.

Example 3:

A spherical ball has a surface area of 2464 sq. cm. Find the radius of the ball, correct to 2 decimal places(using pi = 22/7)

Solution:

Step 1:

Surface area of a sphere = 2464 cm2

To find: Radius of a sphere

Step 2:

Surface area a sphere (SA) = 4 π r2

Radius of a sphere (r) = \(\sqrt{\frac{Surface \ Area}{4\pi}}\)

r = \(\sqrt{\frac{2464 \times 7}{4 \times 22}}\)

= 196

Hence the radius of a sphere is 196 cm.

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