Volume calculations are useful in a wide range of contexts. Here are some applications where finding the volume of a sphere would be useful:
- A toy manufacturer making super bouncy rubber balls would need to know the volume of the ball for manufacturing and packaging of the product.
- An astronomer who wants to calculate how much the sun weighs would need to know its volume.
- A pharmaceutical company making round pills needs to know the dosage of medicine in each pill, which is found using its volume.
- A party equipment hire company needs to know how much gas fits inside each of their balloons, which is found using its volume.
The volume of a sphere with radius $r$r can be calculated using the following formula:
| Volume of a sphere | $=$= | $\frac{4}{3}\times\pi\times r^3$43×π×r3 |
| $=$= | $\frac{4}{3}\pi r^3$43πr3 |
If we are asked to find the volume of a hemisphere, we would find the volume of a sphere with the same radius, then halve the result.
Worked example
Example 1
Calculate the capacity, to the nearest litre, of a hemispherical bird bath with an internal radius of $0.5$0.5 m.
Think: To calculate the capacity we first need to work out the volume.
The volume of a sphere is given by $V=\frac{4}{3}\pi r^3$V=43πr3. Therefore the volume of a hemisphere is a half of this. Note that we are assuming that the bird bath is full of water. The radius of the hemisphere is $0.5$0.5 m, which is $50$50 cm.
| Volume of hemisphere | $=$= | $\frac{1}{2}\times\text{volume of sphere }$12×volume of sphere |
| $=$= | $\frac{1}{2}\times\frac{4}{3}\times\pi\times50^3$12×43×π×503 | |
| $=$= | $261799.3878$261799.3878... cm3 | |
| Since $1$1 L | $=$= | $1000$1000 cm3 |
| Capacity of hemisphere | $=$= | $\frac{261799.2878}{1000}$261799.28781000 |
| $=$= | $261.7993$261.7993... L | |
| $=$= | $262$262 L (nearest litre) |
Practice questions
Question 1
Find the volume of the sphere shown.
Round your answer to two decimal places.
Question 2
A sphere has a radius $r$r cm long and a volume of $72\pi$72π cm3. Find the value of $r$r.
Round your answer to two decimal places.
QUESTION 3
Copper weighs approximately $9$9 grams per cubic centimetre. What is the weight of $8$8 solid spheres of copper having a diameter of $16$16 cm?
Give your answer in grams.
Round your answer to the nearest gram.