How To Find Surface Area Of A Sphere

 

Title:- How To Find Surface Area Of A Sphere

Introduction:-

Today in this article I will try to explain the topic How to Find Surface Area Of A Sphere.
To start this explanation, let us first know some basic knowledge on sphere. Our basic knowledge starts with
What is sphere?
The sphere, a perfectly symmetrical three-dimensional object, can be explained further that sphere is the summation of all points that are all at the same distance from a fixed given point in a three dimensional space. It has fascinated mathematicians, scientists, and thinkers for centuries. Its inherent simplicity and elegance have made it a fundamental shape in various fields of study, including mathematics, physics, and engineering. One of the fundamental properties of a sphere is its surface area, which holds importance in diverse applications ranging from geometry to real-world scenarios and also explore the concept How To Find The Surface Area Of A Sphere, its significance, and the step-by-step methods to calculate it.

Explanation:-

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The Significance of Sphere's Surface Area:
Before diving into the difficulties of calculating the surface area of a sphere, let's understand why this geometric property is significant. The surface area represents the total amount of space occupied by the exterior of the sphere. This property has implications in various real-world contexts, such as calculating heat transfer in thermodynamics, designing spherical objects in engineering, and understanding the behavior of celestial bodies like planets. From soap bubbles to planets, the concept of surface area plays a vital role in understanding the behavior of objects that exhibit spherical symmetry.

The Formula for Surface Area of sphere:-

The formula How To Find Surface Area Of A Sphere is derived from its defining characteristics. A sphere is defined as the set of all points in space equidistant from a given point called the center. The distance from the center to any point on the sphere is referred to as the radius (denoted as "r"). With this definition in mind, the formula for the surface area of a sphere is:
A = 4πr²
 Where:
A represents the surface area of the sphere.
π (pi) is a mathematical constant approximately equal to 3.14159 or 22/7
r is the radius of the sphere.
This formula provides a straightforward and elegant way to compute the surface area of any sphere, regardless of its size.

Step-by-Step Calculation:

Let's break down the process of calculating the surface area of a sphere into simple steps:
(1) Measure the Radius:- Let,us first start by measuring the radius of the sphere. The radius is the distance from the center of the sphere to any point on its surface. We will ensure that the unit of measurement is consistent with our calculations.
(2) Square the Radius:- Once we have the radius value, we will square it by multiplying it by itself. This step is essential for computing the surface area formula.
(3) Multiply by 4 and π:- We will multiply the squared radius by both 4 and the value of π. This step combines the mathematical constants and the squared radius to yield the surface area of the sphere ,A = 4πr²

(4) Finalize the Calculation:- After performing the multiplication, we'll have the surface area of the sphere in square units. We must be sure to include the appropriate unit of measurement (e.g., square inches, square centimeters) based on your initial radius measurement.

Example Calculation:-

Let's illustrate the calculation with an example:
Suppose we have a sphere with a radius (r)  =5 units.
∴ r² = 5² = 25 
Now,
         4π
= 4  × 3.14159 [ ∵ π = 3.14159 is taken]
∴ Area of the sphere
  = 4πr²
 = 4×3.14159×25
 = 314.159 square units.
The surface area of the sphere is approximately 314.159 square units.

Some Problems on Area of sphere:-

Question(1):- The radius of a sphere is 7cm,find its surface area.(Take  π = 22/7)

Answer:-

           Radius(r) = 7cm
∴ Surface area of the sphere = 4πr²
                                  = 4× 22/7 ×(7)²
                                  = 4× 22/7 × 7 × 7
                                  = 4 × 22 × 7
                                  = 616 cm²

Question(2):- Find the total surface area of the hemisphere whose diameter is 14cm.(Take  π = 22/7)

Answer:-

Radius of the hemisphere (r) = 14/2 = 7cm
∴Total Surface Area of the hemisphere = 3πr²
              = 3 × 22/7 × (7)²
              = 3 × 22/7 × 7 × 7[Puttig π = 22/7)
              = 3 × 22 × 7
              = 462 cm²

Question(3):- The total surface area of a sphere is 12.56 m², find the radius of the sphere.(Take π = 22/7)

Answer:- 1st Method,
Total surface area of the sphere (4πr²) = 12.56 m²
Let,
Radius of the sphere = r m
∴ r² =Total surface area of a sphere /4π
=> r² = 12.56 / (4 × 3.14)[∵putting π = 22/7]
=> r² = 4 /4
=> r² = 1
=>r =±1
∴ r = 1 [neglecting - 1 as radius of a sphere never be a negative number]
∴ Radius of the sphere (r) = 1m

2nd Method,
Let,
Radius of the sphere = r m
Now, according to question,
4πr² = 12.56
=> r² = 12.56 / (4 × 3.14) [ ∵ putting π = 3.14 ]
=> r² = 4/4
=> r² = 1
=>r =±1

∴ r = 1 [neglecting - 1 as radius of a sphere never be a negative number]
∴ Radius of the sphere (r) = 1m

Question(4):- If the radius of a sphere is increased by 2 times ,then what will be the ratio between it's original surface area to its new area?
Answer:-
   Let,
          the original radius of the sphere is = r
      and it's new radius after increase(R) = 2r
  ∴ Original surface area(S.A) = 4πr²
and new surface area(S.A₁) = 4πR²
                                       = 4π(2r)²[Putting R=2r]
                                       = 4π4r²
                                       =4πr² ×4
      
∴ S.A              4πr²
          ----------   = ------------
            S.A₁          4πr² ×4
            S.A             1
      => --------- =    ----
            S.A₁            4
∴ Ratio between it's S.A to the new S.A = 1: 4

Conclusion:-

The surface area of a sphere is a fundamental geometric property that holds significance in various fields and real-world applications. Its elegance lies in its simplicity, encapsulated by the formula A = 4πr²
 , where A is the surface area,π is the mathematical constant, and rr is the radius of the sphere. The step-by-step process of calculating the surface area involves squaring the radius, multiplying by mathematical constants, and finalizing the unit of measurement.
Whether we're exploring the behavior of celestial bodies, designing objects with spherical symmetry, or investigating heat transfer phenomena, understanding and calculating How To Find Surface Area Of A Sphere is an essential skill. It provides insights into the spatial characteristics of this captivating...........

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FREQUENTLY ASKED QUESTIONS on Surface Area Of A Sphere:- 

(i).Find the surface area of a sphere of radius 7cm

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Answer:Radius ( r ) = 7cm

∴ Surface area of the sphere (S.A) = 4πr²

= 4 × 22/7 ×(7)²

= 4 × 22/7× 7 × 7

= 4 × 22 × 7

= 616 cm²

(ii).What is TSA of sphere and CSA of sphere?

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Answer:T.S.A = C.S.A of a sphere = 4πr²

(iii).What is the CSA of a circle?

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Answer:A circle is not a three dimensional shape,so it has no C.S.A

(iv).What is a half sphere called?

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Answer:A half sphere called hemi sphere.

(v).What is area of sector class 10?

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Answer:Area of sector = (θ/360°) ×πr² where, θ = Angle of the sector and r = Radius of the circle

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