How to Use the Sphere Formula with Definition and Solved Examples
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 for Surface Area and Volume
1. What is the formula for the volume of a sphere?
The formula for the volume of a sphere is V = (4/3)πr³, where r is the radius of the sphere.
- r = radius (distance from center to surface)
- π ≈ 3.1416
- V = (4/3) × π × 3³
- V = (4/3) × π × 27
- V = 36π ≈ 113.1 cm³
2. What is the surface area formula of a sphere?
The surface area of a sphere is given by A = 4πr², where r is the radius.
- r = radius
- π ≈ 3.1416
- A = 4 × π × 5²
- A = 4 × π × 25
- A = 100π ≈ 314.16 m²
3. How do you calculate the radius of a sphere from its volume?
To find the radius of a sphere from volume, use r = ∛(3V / 4π).
- Start with V = (4/3)πr³
- Rearrange to r³ = 3V / 4π
- Take cube root: r = ∛(3V / 4π)
- r³ = (3 × 288π) / 4π = 216
- r = 6 cm
4. How do you calculate the diameter of a sphere?
The diameter of a sphere is calculated using d = 2r, where r is the radius.
- Measure or find the radius
- Multiply by 2
- d = 2 × 7
- d = 14 cm
5. Why is the volume of a sphere (4/3)πr³?
The volume of a sphere is (4/3)πr³ because it is derived using integral calculus or by comparing a sphere to a cylinder and cone of the same radius and height.
- Archimedes showed that sphere volume = 2/3 of a cylinder's volume.
- Cylinder volume = πr²(2r) = 2πr³
- 2/3 × 2πr³ = (4/3)πr³
6. What is the difference between the surface area and volume of a sphere?
The surface area measures the outer covering of a sphere, while the volume measures the space inside it.
- Surface area formula: 4πr²
- Volume formula: (4/3)πr³
- Surface area uses square units (cm², m²)
- Volume uses cubic units (cm³, m³)
7. How do you find the surface area of a sphere from its diameter?
To find the surface area from diameter, use A = πd², where d is the diameter.
- Since r = d/2, substitute into 4πr²
- 4π(d/2)² = πd²
- A = π × 10²
- A = 100π ≈ 314.16 cm²
8. What are the units used in sphere formulas?
The units in sphere formulas depend on whether you calculate surface area or volume.
- Surface area (4πr²) → square units like cm², m²
- Volume ((4/3)πr³) → cubic units like cm³, m³
9. Can you give a solved example of a sphere formula problem?
Yes, for a sphere with radius 4 cm, the volume is (256/3)π ≈ 268.08 cm³ and the surface area is 64π ≈ 201.06 cm².
- Volume: V = (4/3)π(4³) = (4/3)π(64) = (256/3)π
- Surface area: A = 4π(4²) = 4π(16) = 64π
10. What are common mistakes when using the sphere formula?
Common mistakes in using the sphere formula include confusing radius with diameter and forgetting to cube the radius in the volume formula.
- Using diameter instead of radius without dividing by 2
- Writing volume as 4πr³ instead of (4/3)πr³
- Forgetting to cube r in volume calculation
- Using wrong units (not cubing units for volume)
