The surface area of a sphere is the region covered by its outer surface. On the other hand, the volume represents the three-dimensional space occupied by the figure. We can calculate the surface area of a sphere using the formula A=4πr² and we can calculate its volume using the formula V=(4/3)πr³, where r is the radius of the sphere.
In this article, we will learn all about the area and volume of a sphere. We will explore their formulas and use them to solve some practice problems.
How to find the surface area of a sphere?
We can calculate the surface area of a sphere by multiplying the product of pi and the square of the radius of the sphere by 4. Therefore, the formula for the surface area of a sphere is given by:
| $latex A_{s}=4\pi{{r}^2}$ |
where, $latex A_{s}$ represents the surface area of the sphere and r represents the length of the radius.
Calculate the surface area of a sphere using the diameter
If we know the length of the diameter, we can calculate its surface area using two main methods. The first method consists in dividing the length of the diameter by 2 and using the standard formula for the surface area of a sphere.
The second method consists in finding a formula for the surface area of a sphere in terms of the diameter. Therefore, substituting the expression r=d/2 in the surface area formula, we have:
$latex A_{s}=4\pi{{r}^2}$
$latex A_{s}=4\pi(\frac{d}{2})^2$
$latex A_{s}=4\pi(\frac{d^2}{4})$
| $latex A_{s}=\pi{{d}^2}$ |
How to find the volume of a sphere?
We can calculate the volume of a sphere using the following formula:
| $latex V=\frac{4}{3}\pi {{r}^3}$ |
where, r is the length of the radius of the sphere.
The formula for the volume of a sphere can be proved using integral calculus.
Calculate the volume of a sphere using the diameter
To calculate the volume of a sphere using its diameter, we can use two different methods. The first method consists in dividing the diameter by 2 to get the radius and using the standard formula for the volume of a sphere.
The second method consists in obtaining a formula for the volume of a sphere in terms of the diameter. We can accomplish this by substituting the expression r=d/2 into the volume formula. Therefore, 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.
Calculate the volume of a hollow sphere
We can calculate the volume of a hollow sphere by subtracting the volume of the hollow part from the total volume of the sphere. Therefore, if we use $latex r_{1}$ to represent the radius of the entire sphere and $latex r_{2}$ to represent the internal radius, that is, the radius of the hollow part, we have:
$latex V=\frac{4}{3}\pi {{r_{1}}^3}-\frac{4}{3}\pi {{r_{2}}^3}$
Simplifying, we can obtain the following formula:
| $latex V=\frac{4}{3}\pi ({{r_{1}}^3}-{{r_{2}}^3})$ |
Surface area and volume of a sphere - Examples with answers
The formulas for the surface area and volume of a sphere are used to solve the following examples. Try to solve the problems yourself before looking at the solution.
EXAMPLE 1
Find the surface area of a sphere that has a radius of 4 inches.
Solution
Using the surface area formula with length $latex r=4$, we have:
$latex A_{s}=4\pi {{r}^2}$
$latex A_{s}=4\pi {{(4)}^2}$
$latex A_{s}=4\pi (16)$
$latex A_{s}=201.1$
The surface area is equal to 201.1 in².
EXAMPLE 2
Find the volume of a sphere that has a radius of 3 ft.
Solution
Using the formula for volume in terms of radius with $latex r=3$, we have:
$latex V=\frac{4}{3}\pi {{r}^3}$
$latex V=\frac{4}{3}\pi {{(3)}^3}$
$latex V=\frac{4}{3}\pi (27)$
$latex V=113.1$
The volume is equal to 113.1 ft³.
EXAMPLE 3
What is the surface area of a sphere that has a radius of 5 yards?
Solution
We use the radius $latex r=5$ in the surface area formula. Therefore, we have:
$latex A_{s}=4\pi {{r}^2}$
$latex A_{s}=4\pi {{(5)}^2}$
$latex A_{s}=4\pi (25)$
$latex A_{s}=314.2$
The surface area is equal to 314.2 yd².
EXAMPLE 4
What is the volume of a sphere that has a radius of 4 inches?
Solution
Using the formula for volume in terms of radius with the length $latex r=4$, we have:
$latex V=\frac{4}{3}\pi {{r}^3}$
$latex V=\frac{4}{3}\pi {{(4)}^3}$
$latex V=\frac{4}{3}\pi (64)$
$latex V=268.1$
The volume is equal to 268.1 in³.
EXAMPLE 5
Find the surface area of a sphere that has a diameter of 12 feet.
Solution
Since we have the length of the diameter, we can divide it by 2 to get the radius. Therefore, we use the surface area formula with length $latex r=6$:
$latex A_{s}=4\pi {{r}^2}$
$latex A_{s}=4\pi {{(6)}^2}$
$latex A_{s}=4\pi (36)$
$latex A_{s}=452.4$
The surface area is equal to 452.4 ft².
EXAMPLE 6
Find the volume of a sphere that has a diameter of 5 yards.
Solution
Since we have the diameter of the sphere, we can divide it by 2 to get the radius. This means that the radius is equal to $latex r=2.5$. Therefore, we have:
$latex V=\frac{4}{3}\pi {{r}^3}$
$latex V=\frac{4}{3}\pi {{(2.5)}^3}$
$latex V=\frac{4}{3}\pi (6.25)$
$latex V=26.2$
The volume is equal to 26.2 yd³.
EXAMPLE 7
What is the radius of a sphere that has a surface area of 200 in²?
Solution
In this case, we know the surface area, and we want to find the length of the radius. Therefore, we'll use the surface area formula and solve for r:
$latex A_{s}=4\pi {{r}^2}$
$latex 200=4\pi {{r}^2}$
$latex 50=\pi {{r}^2}$
$latex 15.92={{r}^2}$
$latex r=3.99$
The length of the radius is 3.99 in.
EXAMPLE 8
Find the volume of a hollow sphere that has an external radius of 6 inches and an internal radius of 4 inches.
Solution
The hollow sphere has radii $latex r_{1}=6$ and $latex r_{2}=4$. Therefore, we can use the formula for the volume of a hollow sphere 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 in³.
EXAMPLE 9
Find the radius of a sphere that has a surface area of 460 ft².
Solution
Let's use the surface area formula and solve for r:
$latex A_{s}=4\pi {{r}^2}$
$latex 460=4\pi {{r}^2}$
$latex 115=\pi {{r}^2}$
$latex 36.6={{r}^2}$
$latex r=6.05$
The length of the spoke is 6.05 ft.
EXAMPLE 10
What is the volume of a hollow sphere that has an external radius of 5 inches and an internal radius of 4 inches?
Solution
The radii of the hollow sphere are $latex r_{1}=5$ and $latex r_{2}=4$. Therefore, we have:
$latex V=\frac{4}{3}\pi ({{r_{1}}^3}-{{r_{2}}^3})$
$latex V=\frac{4}{3}\pi ({{(5)}^3}-{{(4)}^3})$
$latex V=\frac{4}{3}\pi (125-64)$
$latex V=\frac{4}{3}\pi (152)$
$latex V=255.5$
The volume is equal to 255.5 in³.
Surface area and volume of a sphere - Practice problems
Use the formulas for the surface area and volume of a sphere to solve the following problems. Click "Check" to make sure if your answer is correct.
See also
Interested in learning more about the area and volume of geometric figures? Take a look at these pages:
