Volume of a Sphere
- Introduction
- What is the Volume of a Sphere?
- Volume of the Sphere Formula
- Derivation of Volume of Sphere
- The Volume of a Solid Sphere
- The Volume of a Hollow Sphere
- Calculating the Volume of a Sphere
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
The volume of a sphere is fundamental in mathematics and various scientific disciplines. The volume of a sphere refers to the amount of space enclosed by the spherical shape. It can be calculated using a simple formula using the sphere's radius.
What is the Volume of a Sphere?
The volume of a sphere is essentially the amount of space that can be occupied by any spherical shape. Imagine drawing a circle on a flat surface, then taking a circular disc, attaching a string along its diameter, and rotating it around that string. The resulting shape is a sphere. The unit used to measure the volume of a sphere is typically given as `(\text{unit})^3`, where the unit could be meters, centimeters, inches, or feet, depending on the system of measurement being used.
Volume of the Sphere Formula
The formula to calculate the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \), where \( V \) represents the volume and \( r \) represents the radius of the sphere. This formula allows us to determine the amount of space enclosed within a sphere based on its radius.
Derivation of Volume of Sphere
According to Archimedes, if we have a cylinder, a cone, and a sphere, all with the same radius \( r \) and sharing the same cross-sectional area, their volumes are related in a specific ratio: `1:2:3`. From this, we can derive a relationship between the volumes of the cylinder, cone, and sphere.
We start with the formula for the volume of a cylinder, which is \( V_{\text{cylinder}} = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. Now, the volume of a cone is one-third of the volume of the cylinder, so \( V_{\text{cone}} = \frac{1}{3} \pi r^2 h \).
Using these formulas, we can express the volume of the sphere in terms of the cylinder and cone volumes. By subtracting the volume of the cone from the volume of the cylinder, we get the volume of the sphere: \( V_{\text{sphere}} = V_{\text{cylinder}} - V_{\text{cone}} \).
Now, substituting the formulas for cylinder and cone volumes, we have \( V_{\text{sphere}} = \pi r^2 h - \frac{1}{3} \pi r^2 h \).
Simplifying this, we find that \( V_{\text{sphere}} = \frac{2}{3} \pi r^2 h \).
In this case, the height of the cylinder is the same as the diameter of the sphere, which is \( 2r \). Therefore, we can rewrite the equation as \( V_{\text{sphere}} = \frac{2}{3} \pi r^2 (2r) = \frac{4}{3} \pi r^3 \).
The Volume of a Solid Sphere
So, the volume of a solid sphere is \( \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the sphere. This formula is widely used in various fields of science and engineering, as it allows us to determine the amount of space occupied by spherical objects and to make calculations related to their properties and interactions.
The Volume of a Hollow Sphere
The volume of a hollow sphere is the amount of space enclosed between the outer and inner surfaces of a hollow sphere. To calculate the volume of a hollow sphere, we subtract the volume of the inner sphere from the outer volume sphere.
If the outer radius of the hollow sphere is \( R \) and the inner radius is \( r \) (where \( R > r \)), the formula for the volume of the hollow sphere is:
\( V = \frac{4}{3} \pi R^3 - \frac{4}{3} \pi r^3 \)
We can simplify this to:
\( V = \frac{4}{3} \pi (R^3 - r^3) \)
Calculating the Volume of a Sphere
`1`. Understand the formula: The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \), where \( V \) represents the volume and \( r \) represents the radius of the sphere.
`2`. Measure the radius: Begin by measuring the radius of the sphere. The radius is the distance from the center of the sphere to any point on its surface.
`3`. Plug the radius into the formula: Once you have the radius value, plug it into the formula \( V = \frac{4}{3} \pi r^3 \).
`4`. Calculate: Perform the necessary calculations. First, cube the radius (multiply it by itself three times), then multiply by \( \frac{4}{3} \pi \).
`5`. Get the Volume: After completing the calculations, you will have the volume of the sphere in cubic units.
Example: Calculate the volume of a sphere with a radius of \( r = 3 \) units. Express the volume in terms of `pi`.
Solution:
Using the formula for the volume of a sphere \( V = \frac{4}{3} \pi r^3 \), we can plug in the given radius:
\( V = \frac{4}{3} \pi (3)^3 \)
Now, let's perform the calculations step by step:
`1`. Cube the radius: \( 3^3 = 27 \).
`2`. Multiply by \( \frac{4}{3} \pi \): \( \frac{4}{3} \pi \times 27 \).
`3`. Calculate: \( \frac{4}{3} \times 27 \times \pi = 36 \pi \) cubic units.
Therefore, the volume of the sphere with a radius of `3` units is \( 36 \pi \) cubic units.
Solved Examples
Example `1`. Find the volume of a solid sphere with a radius of `6` centimeters. Consider \( \pi \ = 3.14 \).
Solution:
Given: Radius (\( r \)) `= 6` `cm`
Using the formula for the volume of a solid sphere: \( V = \frac{4}{3} \pi r^3 \)
Substitute the radius: \( V = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288 \pi\ = 904.32\) cubic centimeters.
The volume of the solid sphere is \( 288 \pi \) cubic centimeters.
Example `2`. A hollow sphere has an outer radius of `10` meters and an inner radius of `8` meters. Calculate the volume of the hollow sphere. Write your answer in terms of \( \pi \).
Solution:
Given: Outer Radius (\( R \)) `= 10` `m`, Inner Radius (\( r \)) `= 8` `m`
Using the formula for the volume of a hollow sphere: \( V = \frac{4}{3} \pi (R^3 - r^3) \)
Substitute the radii: \( V = \frac{4}{3} \pi (10^3 - 8^3) \)
\( V = \frac{4}{3} \pi (1000) - \frac{4}{3} \pi (512) \)
\( V = \frac{4}{3} \pi (1000 - 512) \)
\( V = \frac{1952}{3} \pi \) cubic meters.
The volume of the hollow sphere is \( \frac{1952}{3} \pi \) cubic meters.
Example `3`. A spherical balloon has a volume of \( 500 \pi \) cubic inches. Calculate its radius.
Solution:
Given: Volume (\( V \)) `=` \( 500 \pi \) cubic inches
Using the formula for the volume of a solid sphere: \( V = \frac{4}{3} \pi r^3 \)
Rearrange the formula to solve for \( r \): \( r = \sqrt[3]{\frac{3V}{4\pi}} \)
Substitute the volume: \( r = \sqrt[3]{\frac{3 \times 500 \pi}{4\pi}} \)
\( r = \sqrt[3]{\frac{1500}{4}} \)
\( r = \sqrt[3]{375} \)
\( r \approx 7.28 \) inches (rounded to two decimal places).
The radius of the spherical balloon is approximately `7.28` inches.
Example `4`. The total surface area of a sphere is \( 144\pi \) square units. Find the total volume of the sphere in terms of `pi`.
Solution:
Given: Total surface area (\( A \)) `=` \( 144\pi \) square units
We know that the formula for the total surface area (\( A \)) of a sphere is given by:
\( A = 4\pi r^2 \)
Given the total surface area, we can rearrange the formula to solve for the radius (\( r \)):
\( r = \sqrt{\frac{A}{4\pi}} \)
Substitute the given surface area:
\( r = \sqrt{\frac{144\pi}{4\pi}} = \sqrt{36} = 6 \)
Now that we have the radius (\( r \)), we can use it to find the volume (\( V \)) of the sphere using the formula:
\( V = \frac{4}{3}\pi r^3 \)
Substitute the radius:
\( V = \frac{4}{3}\pi (6)^3 = \frac{4}{3}\pi (216) = 288\pi \) cubic units.
Therefore, the total volume of the sphere is \( 288\pi \) cubic units.
Example `5`. Three spheres of radius `4` `cm` fit inside a tube. Calculate the volume of the tube that is not filled. Consider \( \pi \ = 3.14 \).
Solution:
Solution:
In order to find the unfilled volume of the tube, we need to find the difference between the volume of the cylindrical tube and the combined volume of the `3` spheres.
We know that the formula for the volume of a cylinder (\( V_1 \)) of a sphere is given by:
\( V = \pi r^2 h \),
where \(h \) is the height of the cylinder
\(r \) is the radius of the base
Note:
- The height of the cylinder is `3` times the diameter of each sphere `=` \(3 \times 8\)
Hence `h = 24` `cm`. - The radius of the base of the cylinder is equal to the radius of each sphere `= 4 cm`.
Hence `r = 4` `cm`
Substituting `h = 24` and `r = 4` into the formula \( V_1= \pi r^2 h \), we get the volume of the cylinder as
\( V_1 = \pi \times4^2 \times 24 \)
\( V_1 = \pi \times16 \times 24 \)
\( V_1 = 384 \pi\ cm^3 \)
Substituting `r = 4` into the formula \( V_2 = \frac{4}{3}\pi r^3 \), we get the volume of each sphere as
\( V_2 = \frac{256}{3}\pi\)
Hence the combined volume of the `3` spheres \(=3 \times \frac{256}{3}\pi = 256\pi\ cm^3\)
Unfilled volume of the tube `=` Volume of the cylindrical tube `-` Combined volume of the `3` spheres
Unfilled volume of the tube `=` \( 384 \pi\ cm^3 - 256 \pi\ cm^3\)
Unfilled volume of the tube `=` \( 128 \pi\ cm^3\) `=` \( 401.92\ cm^3\)
Practice Problems
Q`1`. What is the volume of a sphere with a radius of `5` meters? Give the answer in terms of `pi`.
- \( \frac{256}{3}\pi \) cubic meters
- \( \frac{500}{3}\pi \) cubic meters
- \( \frac{625}{3}\pi \) cubic meters
- \( \frac{1000}{3}\pi \) cubic meters
Answer: b
Q`2`. A hollow sphere has an outer radius of `12` centimeters and an inner radius of `8` centimeters. What is the volume of the hollow space? Give the answer in terms of `pi`.
- \( \frac{128}{3}\pi \) cubic centimeters
- \( \frac{1912}{3}\pi \) cubic centimeters
- \( \frac{4864}{3}\pi \) cubic centimeters
- \( \frac{384}{3}\pi \) cubic centimeters
Answer: c
Q`3`. The volume of a sphere is \( 288 \pi \) cubic inches. What is its radius?
- `3` inches
- `4` inches
- `5` inches
- `6` inches
Answer: d
Q`4`. The total surface area of a sphere is \( 100\pi \) square units. Find the total volume of the sphere in terms of `pi`.
- \( \frac{384}{3}\pi \) cubic meters
- \( \frac{725}{3}\pi \) cubic meters
- \( \frac{500}{3}\pi \) cubic meters
- \( \frac{800}{3}\pi \) cubic meters
Answer: c
Q`5`. A spherical tank has a volume of \( 8000 \) cubic feet. What is its diameter, rounded to the nearest whole number.
- `20` feet
- `25` feet
- `28` feet
- `32` feet
Answer: b
Frequently Asked Questions
Q`1`. What is the formula for the volume of a sphere?
Answer: The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \), where \( V \) represents the volume and \( r \) represents the radius of the sphere.
Q`2`. How do I calculate the volume of a sphere if I only know its diameter?
Answer: If you only have the diameter (\( d \)) of the sphere, you can calculate the radius (\( r \)) by dividing the diameter by `2` (\( r = \frac{d}{2} \)), and then use the formula \( V = \frac{4}{3} \pi r^3 \) to find the volume.
Q`3`. Can I find the volume of a hollow sphere?
Answer: Yes, the volume of a hollow sphere can be calculated by subtracting the volume of the inner sphere from the volume of the outer sphere. The formula for the volume of a hollow sphere is \( V = \frac{4}{3} \pi R^3 - \frac{4}{3} \pi r^3 \), where \( R \) is the outer radius and \( r \) is the inner radius.
Q`4`. What are the units for the volume of a sphere?
Answer: The volume of a sphere is typically measured in cubic units, such as cubic meters `(\text{m}^3)`, cubic centimeters `(\text{cm}^3)`, cubic inches `("in"^3)`, or cubic feet `(\text{ft}^3)`, depending on the system of measurement being used.
Q`5`. How can I use the volume of a sphere in real-life applications?
Answer: The volume of a sphere is used in various fields, including mathematics, physics, engineering, and astronomy. It is employed in calculating the capacities of spherical containers, understanding buoyancy in fluids, modeling planetary bodies, and designing objects with spherical components, among other applications.