The volume of a sphere is equal to the space occupied by the sphere in all three dimensions. The volume of a sphere depends on its radius since, if we take the cross-section of the sphere, we have a circle. Therefore, we have to use the length of the radius to calculate its volume. Alternatively, we can also calculate its volume using the diameter since it is simply twice the radius.
Here, we will learn about the formulas that we can use to calculate the volume of spheres using both their radius and their diameter. In addition, we will also learn to calculate the volume of hollow spheres. Finally, we will apply all the learned formulas to solve some problems.
Formula to find the volume of a sphere
The sphere is defined as a solid three-dimensional figure, in which each point on its surface is equidistant from the center. The fixed distance is called the radius of the sphere and the fixed point is called the center of the sphere.
The volume of a sphere is determined using the length of the radius and the following formula:
| $latex V=\frac{4}{3}\pi {{r}^3}$ |
where r is the length of the radius of the sphere.
This formula is derived using integration methods.
Volume of a sphere using the diameter
Alternatively, we can use the diameter of a sphere to calculate its volume. We can do this using two different methods.
The first method is to remember that the diameter of a sphere is equal to 2r, where r is the length of the radius of the sphere. Therefore, if we have the length of the diameter, we can divide by two to get the length of the radius and use the volume formula given above.
The second method is to write the formula for volume in terms of the diameter of the sphere. If we do this, we have:
$latex V=\frac{4}{3}\pi {{r}^3}$
$latex V=\frac{4}{3}\pi {{(\frac{d}{2})}^3}$
$latex V=\frac{4}{3}\pi (\frac{{{d}^3}}{8})$
$latex V=\frac{4}{24}\pi {{d}^3}$
| $latex V=\frac{1}{6}\pi {{d}^3}$ |
where d is the length of the diameter.
Volume of a hollow sphere
The volume of hollow spheres is calculated by subtracting the volume of the hollow part from the volume of the whole sphere. Therefore, if we use $latex r_{1}$ to represent the radius of the complete sphere and $latex r_{2}$ to represent the internal radius, that is, the radius of the hollow part, we have:
$latex V=\frac{4}{3}\pi {{r_{1}}^3}-\frac{4}{3}\pi {{r_{2}}^3}$
We can simplify this to get the following formula:
| $latex V=\frac{4}{3}\pi ({{r_{1}}^3}-{{r_{2}}^3})$ |
Volume of a sphere – Examples with answers
The following examples are solved using the formulas for the volume of spheres seen above. Each example has its respective solution, where the process and reasoning used are detailed.
EXAMPLE 1
What is the volume of a sphere that has a radius of 3 m?
Solution
We use the first volume formula with $latex r=3$:
$latex V=\frac{4}{3}\pi {{r}^3}$
$latex V=\frac{4}{3}\pi {{(3)}^3}$
$latex V=\frac{4}{3}\pi (27)$
$latex V=113.1$
The volume is equal to 113.1 m³.
EXAMPLE 2
A sphere has a radius of 4 m. What is its volume?
Solution
We have to use the first formula of the volume with the length $latex r=4$:
$latex V=\frac{4}{3}\pi {{r}^3}$
$latex V=\frac{4}{3}\pi {{(4)}^3}$
$latex V=\frac{4}{3}\pi (64)$
$latex V=268.1$
The volume is equal to 268.1 m³.
EXAMPLE 3
What is the volume of a sphere that has a diameter of 5 m?
Solution
In this case, we have the diameter of the sphere, so we have to divide by 2 and use the volume formula. This means that the radius is $latex r=2.5$:
$latex V=\frac{4}{3}\pi {{r}^3}$
$latex V=\frac{4}{3}\pi {{(2.5)}^3}$
$latex V=\frac{4}{3}\pi (6.25)$
$latex V=26.2$
The volume is equal to 26.2 m³.
EXAMPLE 4
What is the volume of a hollow sphere that has an external radius of 6 m and an internal radius of 4 m?
Solution
We have a hollow sphere with radii $latex r_{1} = 6$ and $latex r_{2}=4$. Therefore, we use the third volume formula with these radii:
$latex V=\frac{4}{3}\pi ({{r_{1}}^3}-{{r_{2}}^3})$
$latex V=\frac{4}{3}\pi ({{(6)}^3}-{{(4)}^3})$
$latex V=\frac{4}{3}\pi (216-64)$
$latex V=\frac{4}{3}\pi (152)$
$latex V=636.7$
The volume is equal to 636.7 m³.
EXAMPLE 5
A hollow sphere has an external radius of length 5 m and an internal radius of length 4 m. What is its volume?
Solution
The radii of the hollow sphere are $latex r_{1} = 5$ and $latex r_{2} = 4$. Therefore, we use these values in the formula for the volume of a hollow sphere:
$latex V=\frac{4}{3}\pi ({{r_{1}}^3}-{{r_{2}}^3})$
$latex V=\frac{4}{3}\pi ({{(5)}^3}-{{(4)}^3})$
$latex V=\frac{4}{3}\pi (125-64)$
$latex V=\frac{4}{3}\pi (152)$
$latex V=255.5$
The volume is equal to 255.5 m³.
Volume of a sphere – Practice problems
Put into practice the use of the volume formulas seen above to solve the following problems. Select your answer and check it to see that you got the correct answer.
See also
Interested in learning more about spheres? Take a look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
