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Sphere formula- Derivation Volume, Proof, Explaination of Area

Sphere formulas

In geometry, a sphere is described as a geometrical circular object in a 3 three-dimensional plane that has an absolute symmetrical shape. In childhood, we all are used to playing with cricket balls. So, a good example of a sphere is a cricket ball. In geometry, Sphere is a circular shape in a three-dimensional plane. In this article, we will learn about sphere formulas for determining its volume and total surface area.

Sphere formula: What is a Sphere?

A sphere is described as a geometrical circular object in a 3 three-dimensional plane that has an absolute symmetrical shape. All the points  on the surface are present at an equal distance from the center (radius)
Example-Football, tennis ball, globe, etc.

Sphere formula: Total Surface Area (TSA) properties

Center of the sphere-In a sphere all the points on the surface are present at the same distance from a fixed point, the point is regarded as the Center of the sphere.

The radius of the sphere-A straight line which forms a connection between the center of the sphere and any point present on the surface. is known as the radius. Generally, Radius is denoted by the alphabet letter r.

Diameter of a sphere- Diameter is the longest straight line that passes through the center of the sphere and meets any two different points present on the surface of the sphere. Thus. Those two points and the center of the sphere lie in diameter, denoted by D.
In a sphere, the Diameter is always doubled in the radius in length.

Sphere formula- Derivation Volume, Proof, Explanation of Area_40.1
Properties of a sphere

Sphere formula of Area, Diameter, Volume

Diameter of a Sphere D = 2 r
 Area of a Sphere A =4 π r²
 The volume of a Sphere V = 4/3 π r³

Sphere formula Diameter 

Diameter is the longest straight line that passes through the center of the sphere and meets any two different points present on the surface of the sphere. A sphere’s diameter is always doubled because of the radius in length in that particular sphere.
D = 2 r

Sphere formula Area

The total surface area of a sphere is considered as its total curved surface. The sphere does not have any edges or any lateral surfaces. So, we consider that the total surface area of the sphere equals the total curved surface area.
The formula for determining the total surface area of a sphere is,
Total surface area  = 4 π r² square units.
{ In above, π is a constant value which is equal to 3.14 or 22/7 }

Sphere formula- Derivation Volume, Proof, Explanation of Area_50.1
The total surface area of a sphere

Sphere formula Volume

The volume of a Sphere defines as the holding capacity of the sphere into it. The formula for determining the volume of a sphere is –
Let’s assume V is the volume of the sphere,
Then, Volume = 4/3 π r³  cubic units, in which π is a constant that equals 3.14 or 22/7, and r is the radius of the sphere.

Sphere formula- Derivation Volume, Proof, Explanation of Area_60.1
The volume of a sphere

Sphere formula: Difference between Circle and Sphere Explanation

Although the shape of both figures is circular, the main difference between Circle and a sphere is,  Circle is a two-dimensional figure, on the other hand, a sphere is a three-dimensional object. A sphere has area and volume. But a circle only has its area and circumference.

Sphere formula- Derivation Volume, Proof, Explanation of Area_70.1
Difference between a circle and a sphere

Sphere formula Proof with Example

Q. Diameter of a sphere is 12 cm . Find the Radius and total surface area of the sphere.
a) 6 cm and 270 πcm² b) 6 cm and 144 π cm² c)  12 cm and 144 π cm²  d) None of these. 
Answer-Let’s suppose the radius of the sphere is r
The diameter of the given sphere is -D= 12 cm.
We all know that Diameter is doubled in radius in length.
So, D= 2r
or, 12 = 2r
or, r = 6 cm

The formula for the total surface area of a sphere is –
4 π r² square units.
Total surface area= 4 π (6)²
= 4× π ×36
= 144 π or 452.16 cm²

Q. Find the volume of a sphere which have a radius of   3 cm.
a) 27π cm³ b) 36π cm³ c)  36 cm³ d) None of those
Answer-
The formula for determining the volume of the sphere is-
V = 4/3 π r³  cubic units.

Volume- V=  4/3  π  ( 3) ³  cm³
V= 36 π  cm³

So, option ( b) is correct.

Q.If the Surface area of a sphere is 616 cm² . Find the radius of the sphere.
(a) 4  cm   (b) 7 cm (c)   49 cm (d) None of these.
Answer- we know that the formula for the surface area of as phere is – 4πr²
Here, the Surface area of the sphere is 616 cm².
4πr² = 616
r² = 616 /(4 × 22/7) [ π =22/7]
r² = 616 ×7 / 22×4
r² = 49
or, r = 7 cm
The radius of the sphere is 7 cm.

Hence, option (b)is correct.

Q. If the radius of a sphere is increased 2 times. What is its volume? (Radius = r)
(a) 4/3 πr³  (b)43/3πr³  (c)  32/3 r³ (d) None of these.
Answer – The previous radius of the  sphere is  r
After increasing 2 times, the radius is  2r

The volume of the sphere is – 4/3 π(r)³
= 4/3 π(2r)³ = 4/3 π(8r³ ) = 32/3 r³ cubic units .

Hence, option (c)is correct.

Sphere formula-Based Practice questions

  1. If the radius of a sphere is increased 2 times. What is its total surface? (Radius = r)
  2. The diameter of a sphere is 6 cm. Find the Radius and total surface area of the sphere.
  3. Find the volume of a sphere which have a radius of   5 cm.
  4. If the Surface area of a sphere is 343 cm³. Calculate the diameter of the sphere.

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Sphere formula: FAQs

Q. Is the sphere three-dimensional?
Answer- Yes. The sphere is a three-dimensional circular figure.

Q. What is a Sphere?
Answer- A sphere is described as a geometrical circular figure in a 3 three-dimensional plane that has an absolute symmetrical shape.
Example-football ,cricket ball ,globe etc.

Q. How to calculate the volume of a sphere?
Answer- The formula for  calculating the  volume of a sphere is
V = 4/3 π r³  cubic units, in which  π is a constant that equals 3.14 or 22/7,r is the radius of the sphere

Q. How to find the surface area of a sphere?
Answer-  The total surface area of a sphere is considered as its total curved surface.
The formula for determining the total surface area of a sphere is – = 4 π r² square units.

Q. What is the difference between Circle and a sphere?
Answer- Although the shape of both figures is similar, the main difference between Circle and a sphere is  Circle is two-dimensional, on the other hand, the sphere is a three-dimensional figure.

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