Volume of Spheres
The sphere is round and three-dimensional in shape. When a circle is rotated around its diameter, the result is a sphere. The fixed distance from the center of the sphere to any point on its surface is called its radius.
The volume of a sphere can be calculated in terms of its radius. Cubic units, such as \( m^3,~cm^3,~in^3 \), and so on, are used to measure the volume of a sphere.
We come across a number of sphere-shaped objects in our daily life, such as a ball, an orange, the moon, and so on.
List of formulas for the Volume of a Sphere
The volume of a sphere can be calculated using the following formulas:
1. The volume of a sphere, \( V=\frac{4}{3}\times \pi \times r^3 \)
2. The volume of a hollow sphere, \( V=\frac{4}{3}\times \pi\times (R-r)^3\)
3. The volume of a hemisphere, \( V=\frac{2}{3}\pi r^3 \)
These three formulas will be elaborated further.
How can we use the Volume of a Sphere formula?
To calculate the volume of a sphere, we need to measure its radius.
Mathematically,
The volume of a sphere, \( V=\frac{4}{3}\times \pi \times r^3 \) cubic units
Where,
- ‘r’ is the radius of the sphere.
- ‘π’ or Pi, is a mathematical constant and is the ratio of the circumference of a circle to its diameter. The value of π (pi) is \( \frac{22}{7} \), or approximately 3.14.
Hollow Sphere
A hollow sphere is a sphere that has been hollowed in such a way that a wall with a uniform width creates an inner sphere within the outer sphere. The radius of the outer sphere is greater than that of the inner sphere.
The volume of a hollow sphere can be calculated by subtracting the volume of the inner sphere from the volume of the outer sphere.
Mathematically,
The volume of a hollow sphere, \( V=\frac{4}{3}\times \pi \times (R-r)^3 \) cubic units
Where,
- ‘R’ is the radius of the outer sphere
- ‘r’ is the radius of the inner sphere
Hemisphere
If a sphere is cut into two halves with a plane, each half is known as a hemisphere.
The volume of a hemisphere is half of the volume of the full sphere.
Mathematically,
The volume of a hemisphere, \( V=\frac{1}{2}\times \) the volume of the full sphere.
\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ =\frac{1}{2}\times \frac{4}{3}\pi r^3 \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{2}{3}\times \pi r^3 \) cubic units.
Solved Examples
Example 1: Find the volume of a sphere having a radius of 6 inches.
Solution:
The radius of the given sphere is r = 6 inches.
\( V=\frac{4}{3} \pi r^3 \) Formula for the volume of a sphere
\( ~~~=\frac{4}{3} \pi (6)^3 \) Substitute 6 for the radius
\( ~~~=\frac{864}{3} \pi \) Simplify
\( ~~~=904.32 \) Use a calculator
The volume of the sphere will be 904.32 cubic inches.
Example 2: Find the volume of a sphere whose diameter is 8 cm.
Solution:
Given, diameter = 8 cm
So, radius \( =\frac{\text{diameter}}{2}=\frac{8}{2}=4 \) cm
\( V=\frac{4}{3} \pi r^3 \) Formula for the volume of a sphere
\( V=\frac{4}{3} \times \pi \times (4)^3 \) Substitute 4 for the radius
\( V=268.1 \) Use a calculator
Therefore, the volume of the sphere is 268.1 cubic centimeters.
Example 3: Find the volume of a hemisphere whose radius is 3 meters.
Solution:
The given radius of the hemisphere is r = 3 meters.
\( V=\frac{2}{3} \pi r^3 \) Formula for the volume of a hemisphere
\( V=\frac{2}{3} \times \pi \times (3)^3 \) Substitute 3 for the radius
\( V=56.55 \) Use a calculator
Therefore, the volume of the hemisphere is 56.55 cubic meters.
Example 4: Find the volume of a hollow sphere. The outer radius of the sphere is 7 feet and the thickness is 1 foot.
Solution:
The given outer radius of the hollow sphere, R = 7 ft.
The inner radius of the sphere = the outer radius – the thickness
= 7 – 1
= 6 ft
\( V=\frac{4}{3} \pi (R-r)^3 \) Formula for the volume of a hollow-sphere
\( V=\frac{4}{3} \pi (7-6)^3 \) Substitute the value of the radii (R and r)
\( V\approx 4.19 \) Use a calculator
Therefore, the volume of the hollow sphere is about 4.19 cubic feet.
Example 5: An inflated spherical balloon has a radius of 5 feet. Assume that air is leaking from the balloon at a constant rate of 20 cubic feet per minute. How long will it take to deflate the balloon completely?
Solution:
The volume of a spherical balloon \( =\frac{4}{3}\times \pi \times r^3 \)
\( V=\frac{4}{3} \times \pi \times 5^3 \) Substitute the value of the radius
\( V=\frac{500}{3} \pi \) Simplify
\( V\approx 523.3 \) cubic feet Simplify
To calculate the time taken by the balloon to completely deflate, divide the volume of the balloon by the rate of leakage,
The time taken to completely deflate \( =\frac{5233}{20} \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=26.17 \) minutes [Divide]
Therefore, the balloon will completely deflate in about 26.17 minutes.
Example 6: A company creates a hollow sphere with a thickness of 20 cm and an inside radius of 5 m. What will be the volume of the sphere designed by the company?
Solution:
For the given hollow sphere, the inner radius is 5 m and the thickness is 20 cm.
Inner radius, 5 m = 500 cm [Convert m into cm]
So, the outer radius will be the sum of the inner radius and the thickness,
= 500 + 20
= 520 cm
\( V=\frac{4}{3} \times \pi \times (R-r)^3 \) Formula for the volume of a hollow-sphere
\( V=\frac{4}{3} \pi (520-500)^3 \) Substitute the value of the radii (R and r)
\( V\approx 33493.3 \) Use a calculator
Hence, the volume of the sphere designed by the company is \( 33493.3~cm^3 \).
A sphere does not have a base. It only has a curved exterior surface.
The height of a sphere is its diameter. So, in terms of the radius of a sphere, the height of a sphere is twice the radius of the sphere.
The volume of a sphere will be two-third of a cylinder if the height and radius of the cylinder measure the same as the radius of the sphere.
The total surface area, A, of any sphere is given by
A = 4πr2, where ‘r’ stands for the radius of the given sphere.