How to Apply the Sphere Formula in Maths Problems
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
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
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.
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.
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.
FAQs on Sphere Formula Made Simple: Definitions, Examples & FAQs
1. What is a sphere and what is its defining property in three-dimensional space?
A sphere is a perfectly round geometrical object in three-dimensional space. Its defining property is that it consists of all points that are at an equal distance from a fixed central point. This fixed distance is known as the radius (r) of the sphere, and the fixed point is its center.
2. What are the primary formulas used to calculate a sphere's surface area and volume?
The two fundamental formulas for a sphere are based on its radius (r):
- Surface Area (A) = 4πr²: This formula calculates the total area of the outer surface of the sphere.
- Volume (V) = (4/3)πr³: This formula calculates the amount of space enclosed by the sphere.
3. How is a sphere fundamentally different from a circle?
The fundamental difference between a sphere and a circle lies in their dimensions. A circle is a two-dimensional (2D) shape, existing on a flat plane with properties like area and circumference. In contrast, a sphere is a three-dimensional (3D) object with properties like surface area and volume. Essentially, a sphere is the 3D counterpart of a 2D circle.
4. Why does a sphere have only one surface area formula, unlike a cylinder which has both CSA and TSA?
A sphere has only one continuous, curved surface with no flat bases or edges. Therefore, its entire surface is its 'curved surface'. This is why it doesn't have separate formulas for Curved Surface Area (CSA) and Total Surface Area (TSA). The Total Surface Area of a sphere is simply referred to as its surface area (4πr²). A cylinder, on the other hand, has a curved side and two flat circular bases, requiring separate calculations for its curved area and total area.
5. What formulas are used to find the volume and surface areas of a hemisphere?
A hemisphere is exactly half of a sphere. It has a curved surface and a flat circular base. Its formulas are:
- Volume = (2/3)πr³ (half the volume of a sphere).
- Curved Surface Area (CSA) = 2πr² (half the surface area of a sphere).
- Total Surface Area (TSA) = 3πr² (the sum of its curved surface area and the area of its flat circular base, which is πr²).
6. How can you apply the sphere volume formula in a real-world scenario?
A common real-world application is calculating the capacity of a spherical object, like a water tank. For example, to find the volume of water a spherical tank with a radius of 3 metres can hold, you would use the formula V = (4/3)πr³. Substituting the values, the calculation would be V = (4/3) × π × (3)³, which equals 36π cubic metres. This tells you the maximum capacity of the tank.
7. What is the impact on a sphere's surface area and volume if its radius is doubled?
If the radius (r) of a sphere is doubled to (2r):
- The new surface area becomes 4π(2r)² = 4π(4r²) = 16πr². This is four times the original surface area.
- The new volume becomes (4/3)π(2r)³ = (4/3)π(8r³) = (32/3)πr³. This is eight times the original volume.
This shows that changes in the radius have a much greater impact on the volume than on the surface area.
8. What role does Pi (π) play in the sphere formulas?
Pi (π) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. Since a sphere is fundamentally related to a circle (it is a collection of circular cross-sections), π is essential for relating its linear dimensions (like the radius) to its spatial properties (surface area and volume). It acts as a crucial proportionality constant that ensures the formulas accurately describe the sphere's curved nature.
