5.05 Volume of spheres

Lesson

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 sphere

 Volume of a sphere $=$= $\frac{4}{3}\times\pi\times r^3$43​×π×r3 $=$= $\frac{4}{3}\pi r^3$43​πr3

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.

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.

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?